tag:blogger.com,1999:blog-52797635448396100472024-03-12T20:54:29.044-07:00Astronomy & MusicIndependent Research Astronomer and Space Musician
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Come with me and re-discover the universe!
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More info via links on the right.Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.comBlogger170125tag:blogger.com,1999:blog-5279763544839610047.post-12521728347490941812024-03-03T18:42:00.000-08:002024-03-03T18:42:51.623-08:00Sandhill Crane Murmurations on Leap Day 2024<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgk1NZdd7UP7g0p8_3t0RiPNi310A-EPkRIEjNSmrWnallKwqVF-kkGvDunUtXCP3N04tXh0NR7HX8MISGA75netvBZAOS8d0nta7qK5SUzC5MnKyruB469UbYDHvxrIdFCf6FFzNAVcZ7Zw_Sy7Ld2xoaOlzenMP4b8fE5H9m1vBtog4bykHUYNtKAfqm9" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEgk1NZdd7UP7g0p8_3t0RiPNi310A-EPkRIEjNSmrWnallKwqVF-kkGvDunUtXCP3N04tXh0NR7HX8MISGA75netvBZAOS8d0nta7qK5SUzC5MnKyruB469UbYDHvxrIdFCf6FFzNAVcZ7Zw_Sy7Ld2xoaOlzenMP4b8fE5H9m1vBtog4bykHUYNtKAfqm9=w400-h225" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgs3zS9s0sZFxOUIB3me0Dn8m5eZ4NKlnHdqkITNxPO1qiQ6qbzuYh24Z4EANMpf-N13RjPaOLRgr-f-cQIdnav3tCqgGLFUlXkzECoxj-o6JCEGJ6jdG5K4wyab_WCWX3Zn75PV6JC-fw-dP-FkNoPkls8k7ne3xwdzKdq0nRzxujHYc_kuG83s-gx8AN0" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEgs3zS9s0sZFxOUIB3me0Dn8m5eZ4NKlnHdqkITNxPO1qiQ6qbzuYh24Z4EANMpf-N13RjPaOLRgr-f-cQIdnav3tCqgGLFUlXkzECoxj-o6JCEGJ6jdG5K4wyab_WCWX3Zn75PV6JC-fw-dP-FkNoPkls8k7ne3xwdzKdq0nRzxujHYc_kuG83s-gx8AN0=w400-h225" width="400" /></a></div><br /><br /><div><br /></div><div><br /></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-86752643633744311252024-02-29T09:30:00.000-08:002024-02-29T09:30:36.690-08:00Rock Balancing On The Leap Day 2024<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj3Q-eBaA0sxZeHbm_Op5jff1MVkxFdur72e93ns1voFuK3lI5Ay3JTAQowmcNsvGBd0n_eRVFvGHAiGCxeFtbkwcpmphasHbSQNTBuQtNJlDsssUh9niJQvHrI9pWAECrditmDhvy9VWazX36KbgTYfkXeThqAu9frdYJWVkc8M5e6S4O14QVscqe7c2t_" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="816" data-original-width="1092" height="299" src="https://blogger.googleusercontent.com/img/a/AVvXsEj3Q-eBaA0sxZeHbm_Op5jff1MVkxFdur72e93ns1voFuK3lI5Ay3JTAQowmcNsvGBd0n_eRVFvGHAiGCxeFtbkwcpmphasHbSQNTBuQtNJlDsssUh9niJQvHrI9pWAECrditmDhvy9VWazX36KbgTYfkXeThqAu9frdYJWVkc8M5e6S4O14QVscqe7c2t_=w400-h299" width="400" /></a></div><br /><p></p>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-21888325810140444082024-01-22T15:22:00.000-08:002024-01-22T15:22:51.342-08:00Updated Fourier Transform<div style="text-align: justify;">I've updated my FFT code to present spectral data (in either accelerometer data, or sound data, or CA data, or whatever) in the correct way. There was nothing necessarily *wrong* with how I've been doing it, but now I can take full advantage of the temporal aspects of the data. With the old code, I was limited to a temporal resolution proportional to the size of the FFT window. Now I have full control of the time resolution up to the Nyquist frequency. With an improved time resolution, the resulting spectrograms can (and usually will) look much better than before.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The downside to this is that the output data has much larger dimensions. For example, audio data that comes in at 44.1k Hz (normal audio recording), means that one second of data using a 1024 window FFT, results in a spectrogram that's 512 x 44,100 pixels in size! If I have a recording that's five minutes long, that's a spectrogram that's 512 x 13,230,000 pixels in size! I can easily deal with this by not using every data point, but rather every X data points, where X can be interval I want. This effectively widens the time interval between individual spectra, but in actuality there are very few instances where I'm going to need a sampling rate anything higher than maybe 5k Hz. So I can use an 'X' value of 100, and still be measuring data every 0.0022 seconds.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Anyhow, this is more of an internal improvement to the code and not very visible to anyone but me.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Here's 5 seconds of the complex call of a Curve-Billed Thrasher (for which I'm fortunate to have such a creative songbird literally right outside my window year-round). Time goes left to right (0 to 5 seconds), and frequency goes bottom to top (0 to 22.05k Hz):</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjEcA5ac11XmmakqSrs_PoD9UqBgMOnudVuiQmDXfr8MjYEop9oiOL4KjaRIPw9EMnEm8iovK47fH3YwOXao7vtVlXzw6-XZ5lT-zXxLKOlFclCwKAOrehhw0J8X94jVPCZL3Bp46tlfyAjScrMVYQZ7n-tO5xxjASVkyDQjQeNcKJcZy4Roy3uM-Qfm_Km" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="512" data-original-width="1440" height="143" src="https://blogger.googleusercontent.com/img/a/AVvXsEjEcA5ac11XmmakqSrs_PoD9UqBgMOnudVuiQmDXfr8MjYEop9oiOL4KjaRIPw9EMnEm8iovK47fH3YwOXao7vtVlXzw6-XZ5lT-zXxLKOlFclCwKAOrehhw0J8X94jVPCZL3Bp46tlfyAjScrMVYQZ7n-tO5xxjASVkyDQjQeNcKJcZy4Roy3uM-Qfm_Km=w400-h143" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><br /></div>Listening to the sounds and looking at the spectrograms has got me very interested in trying to make sense of these sounds. It's automatic for me: start running the scientific method on this stuff and see where it leads me. LOL like to many other things, probably nowhere!<br /><br /></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-1779307930487395012024-01-01T15:54:00.000-08:002024-01-08T12:26:05.482-08:00Sandhill Cranes Generated by DALL-E3<div class="separator" style="clear: both; text-align: left;">Pretty impressive...</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Prompt: generate a 1794 by 1024 realistic image of a sandhill crane in flight over a playa in southeastern arizona usa</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhPACZW17ar6clBcGlfoNz3TKRtTlmqhH_JDNR23ZyKUxKsALZK9kXv1RkcH_y1LBttQulDQeBo6r-0AJecGVdAPCseAD1FTUFPRQfH2Cmu-u8l6a3uYFIhDV29AILuPD904JcAffNu9PEMGakYjdsYwEeJRPPPYfgV86tURuV6H9vLg1xllZ3XXle2eW-2" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEhPACZW17ar6clBcGlfoNz3TKRtTlmqhH_JDNR23ZyKUxKsALZK9kXv1RkcH_y1LBttQulDQeBo6r-0AJecGVdAPCseAD1FTUFPRQfH2Cmu-u8l6a3uYFIhDV29AILuPD904JcAffNu9PEMGakYjdsYwEeJRPPPYfgV86tURuV6H9vLg1xllZ3XXle2eW-2=w400-h229" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhHeZYp1wyyrFpvvUFTAVp_QeFwEGNmEWuXkny93IguB7GMHS2sROmXVGYnbxBwV8CfPsI9f1czT6IpAYC_rAap3OKvo-R-s0cWckrHM759Ig1Kqafwd-w8th3rkJhZOEsbUmfdxAJRzzqC8XXJc96PdkJAHkmJ-awV7xKWV3eizJ1ZpPrH5IOl5Uk2ybsj" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEhHeZYp1wyyrFpvvUFTAVp_QeFwEGNmEWuXkny93IguB7GMHS2sROmXVGYnbxBwV8CfPsI9f1czT6IpAYC_rAap3OKvo-R-s0cWckrHM759Ig1Kqafwd-w8th3rkJhZOEsbUmfdxAJRzzqC8XXJc96PdkJAHkmJ-awV7xKWV3eizJ1ZpPrH5IOl5Uk2ybsj=w400-h229" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjUoqDRvFXQmX2QgMCJTrI1CbtBbzTbJX6PCwINSSNMtyspAGGmOfK4sDfdYtj_C8HJueYMylEAzn-toRYOtVU9NHXMdLyljYDe9PiMf1xv2xNzj0b5OrWo_uiMbVpp3oxWHbH05znDApQ4FI3z398H2xpQTJHi6M_lAf_oA0TkMjwmQCd__NmMCfcD-IA5" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEjUoqDRvFXQmX2QgMCJTrI1CbtBbzTbJX6PCwINSSNMtyspAGGmOfK4sDfdYtj_C8HJueYMylEAzn-toRYOtVU9NHXMdLyljYDe9PiMf1xv2xNzj0b5OrWo_uiMbVpp3oxWHbH05znDApQ4FI3z398H2xpQTJHi6M_lAf_oA0TkMjwmQCd__NmMCfcD-IA5=w400-h229" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi1KbrD9DU0BQc9Utx1Zd309XIM9jIac6l5Aggj-xxz0DYIX1rcJJfNTfhui9bHBqilij3np55-zHD4zIo67Ys1e9CpmvocGpcG61BAp07Bz96GNOkKTOC7T5voDj_oW05leNOxEm5nNz62k-yDK6L2AGuqq0Ea3owqiMUU1XMrsHvmc884lyNdp6CMXUuB" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEi1KbrD9DU0BQc9Utx1Zd309XIM9jIac6l5Aggj-xxz0DYIX1rcJJfNTfhui9bHBqilij3np55-zHD4zIo67Ys1e9CpmvocGpcG61BAp07Bz96GNOkKTOC7T5voDj_oW05leNOxEm5nNz62k-yDK6L2AGuqq0Ea3owqiMUU1XMrsHvmc884lyNdp6CMXUuB=w400-h229" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">and then a further prompt: generate two more images, but with about five sandhill cranes</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgEM9paMbnlIBKI6Oy-h-ydVkSUCYC-wUNkHUgX91ePtwygaWMu64NuhYvx_vvAXiFeuu7jUKvSjJw_cUv84d_o6iK0g-srrCOORKqyUI1vt7HcNkAMAnBbwGdG2RPuV8AjDDDbwZPUkGHAm9Cz6y5hqubnW8XWgBIk6J4nUkVkiIIl4-qc3raJS_ZebJ1J" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEgEM9paMbnlIBKI6Oy-h-ydVkSUCYC-wUNkHUgX91ePtwygaWMu64NuhYvx_vvAXiFeuu7jUKvSjJw_cUv84d_o6iK0g-srrCOORKqyUI1vt7HcNkAMAnBbwGdG2RPuV8AjDDDbwZPUkGHAm9Cz6y5hqubnW8XWgBIk6J4nUkVkiIIl4-qc3raJS_ZebJ1J=w400-h229" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjTQOh4LWyMwhVRgx6W18BF4JnLPl-oQ_OVIWJEZUTH0YngoLnl6bJcZiTI3KZqpbHUaNrOqEpLVok_NovOqMJymGT_t7pWtEXj5LBvMfYMiLjUhCVn3q0pWHHWzkYA-MjtedKp19d0R_Kv5KtyrYDOipVNSwYKshr3gyaqd2aByhEtTjJHzwuc1DldeigA" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEjTQOh4LWyMwhVRgx6W18BF4JnLPl-oQ_OVIWJEZUTH0YngoLnl6bJcZiTI3KZqpbHUaNrOqEpLVok_NovOqMJymGT_t7pWtEXj5LBvMfYMiLjUhCVn3q0pWHHWzkYA-MjtedKp19d0R_Kv5KtyrYDOipVNSwYKshr3gyaqd2aByhEtTjJHzwuc1DldeigA=w400-h229" width="400" /></a></div><br /><div>Prompt: generate 1794 by 1024 images of a sandhill crane in flight over a playa in southeastern arizona usa. draw it in a way a 5 year old child would with crayon</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgMDUvBg0Vxy5byGpbLD7Z6xjsGTuoVVmBh6RGi-zmEwJVfmHBTdKMS84s4d_RhwAKToHhjSq8P7DA_f3vU2aJWBSUCUmRQqNugjzajwI7cGhTtsK7OJNpxnOm4UKJ8VJncZKG1J5KiMO4Swb51v1nXEjLMdiE1s3dPD0uiTzToxVxyBJn02grJuVU5n31P" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEgMDUvBg0Vxy5byGpbLD7Z6xjsGTuoVVmBh6RGi-zmEwJVfmHBTdKMS84s4d_RhwAKToHhjSq8P7DA_f3vU2aJWBSUCUmRQqNugjzajwI7cGhTtsK7OJNpxnOm4UKJ8VJncZKG1J5KiMO4Swb51v1nXEjLMdiE1s3dPD0uiTzToxVxyBJn02grJuVU5n31P=w400-h229" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhFanopRvkiGitFLoGOJ8Yo1octI-7KWjdtTfQWa2tACA9sBqKaMoqQaKKDfHJyNwS5Zag0BGice1i8r2CFKtRvh882VIn1e14M-X9XN3pkdrcw2uoVHIsruShUAAuq1181WlDXV6CDoXf7XvCeFFimAZHLhylTyA-cFM3KPj80lT9z9PsZV6AGweX741it" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEhFanopRvkiGitFLoGOJ8Yo1octI-7KWjdtTfQWa2tACA9sBqKaMoqQaKKDfHJyNwS5Zag0BGice1i8r2CFKtRvh882VIn1e14M-X9XN3pkdrcw2uoVHIsruShUAAuq1181WlDXV6CDoXf7XvCeFFimAZHLhylTyA-cFM3KPj80lT9z9PsZV6AGweX741it=w400-h229" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Prompt: please make them more simplistic and drawn by a child under the age of five</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjkNqr8jWQzFx4auWB_j4P-BzBVoq5W0RC5OvZzm42iGgKFHPDo8xUuejgXcUD7fF0IPpDemrOvrDhSC7mm5ZqJGYBUq7HqTdN358ygAbtQIQkEXWAI0r85wNlrVTG4GMYWQsUB29Ah14WYGm4dEzQgwidNCKo9AmTti1nqCht1-8_H3BO-COrY7w-xYw-t" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEjkNqr8jWQzFx4auWB_j4P-BzBVoq5W0RC5OvZzm42iGgKFHPDo8xUuejgXcUD7fF0IPpDemrOvrDhSC7mm5ZqJGYBUq7HqTdN358ygAbtQIQkEXWAI0r85wNlrVTG4GMYWQsUB29Ah14WYGm4dEzQgwidNCKo9AmTti1nqCht1-8_H3BO-COrY7w-xYw-t=w400-h229" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhRgDUIFGF4ohxFppJvqXqYbiY6q9vbTObOt7RwsyqKX0rkfv8qBBhM6eZEqZa8BSc5Xpjj6XTXzsKI4loDCkGcN5eZ4smm7fGWRRRmDI0zR5jFnTYpn-Jo7mSiCZNrOhCauL9y7ZlEgXZdzafc5uwcIKQw-IjRoS7_VDI6gNIPBMyxa26n22KVjoD-Y-WG" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEhRgDUIFGF4ohxFppJvqXqYbiY6q9vbTObOt7RwsyqKX0rkfv8qBBhM6eZEqZa8BSc5Xpjj6XTXzsKI4loDCkGcN5eZ4smm7fGWRRRmDI0zR5jFnTYpn-Jo7mSiCZNrOhCauL9y7ZlEgXZdzafc5uwcIKQw-IjRoS7_VDI6gNIPBMyxa26n22KVjoD-Y-WG=w400-h229" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">prompt: please generate two more images in the same style, but draw a small flock of sandhill cranes</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjD_zbAcdZssDHfoxcz7HGLux8npL8GDtodFBagUqy679jPS9xLYB6yCG1FG-v23ThM9J4vJIdC62z7S698VBXidoUu-2VScXaV6Nu-8UFZ9tZu-iyYBEq4Yu0jeOa8SDFMYuzf7B2ODSQHmpdcqkAwBx4sGZvzZT4IiFvzSQG0PyMfPHUEFEgIDIBneaWk" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEjD_zbAcdZssDHfoxcz7HGLux8npL8GDtodFBagUqy679jPS9xLYB6yCG1FG-v23ThM9J4vJIdC62z7S698VBXidoUu-2VScXaV6Nu-8UFZ9tZu-iyYBEq4Yu0jeOa8SDFMYuzf7B2ODSQHmpdcqkAwBx4sGZvzZT4IiFvzSQG0PyMfPHUEFEgIDIBneaWk=w400-h229" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhcV1K_ViXJJTEzw9R2UXLXNdn8Fj-Mwlkc1ZWKZeDOZu64o4e0rC2u7Zpx8WyLVq9LM804VGI0JTT6uYrZt0Y60WnA7hMRo6MsiWpnyUSHDWqTpU__fLeAyvRXaaaGdOdYxedOURL32H2ttDvkMihbtHDWr3cykrMDB4MRoVgVWKmyh2DmMuvQA1eHXgPI" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEhcV1K_ViXJJTEzw9R2UXLXNdn8Fj-Mwlkc1ZWKZeDOZu64o4e0rC2u7Zpx8WyLVq9LM804VGI0JTT6uYrZt0Y60WnA7hMRo6MsiWpnyUSHDWqTpU__fLeAyvRXaaaGdOdYxedOURL32H2ttDvkMihbtHDWr3cykrMDB4MRoVgVWKmyh2DmMuvQA1eHXgPI=w400-h229" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div style="text-align: left;"> I really dig the shadows on that last one. The creation of a 5-year-old child? Sure!</div><br /></div><br /><br /></div></div></div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-63659053204016496312023-12-18T15:47:00.000-08:002023-12-18T16:43:40.833-08:00Bees and BirdsA bee was flying around my camera while I was recording the flights of the sandhill cranes, so i got a pretty decent audio recording of it. I can confirm that a bee beats its wings at about 250 Hz, but wow look at those harmonics!<div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjq5dGLwhJrAmnC3wLx7eEf-khnCgRlMTMKq0OIR3yV3wT2J7l-0JhUTsKBhTuLNlAfzcmUpYgwNyOqzae4Ecx4lhWDDbMyR080CpQJvBS5_HG2FlQhVvp05hYtg5k-dAaxk7AA7Rx7IceujCR65awUAii6QHSyl1-I0N2pI-VP7f8SH00A6FWwsmfumfMo" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="528" data-original-width="563" height="375" src="https://blogger.googleusercontent.com/img/a/AVvXsEjq5dGLwhJrAmnC3wLx7eEf-khnCgRlMTMKq0OIR3yV3wT2J7l-0JhUTsKBhTuLNlAfzcmUpYgwNyOqzae4Ecx4lhWDDbMyR080CpQJvBS5_HG2FlQhVvp05hYtg5k-dAaxk7AA7Rx7IceujCR65awUAii6QHSyl1-I0N2pI-VP7f8SH00A6FWwsmfumfMo=w400-h375" width="400" /></a></div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhW7lcGzMtZ9v4cqHACnWtKHX_JU8vMf8dS4abiio_-PCxuVdSm5J9kAkoSQja4JSgkRLjNhEdtfvYqfBz7uY_qFdejXAHdTNDi4b4Ve7aD9_yudLxNz78UA-PFd3-S7DEvezWRB1MkZDglkMcQc-3aOZ9Nr1Oz7IKod3dUj_tszoFLwvQj3gxAmy_zAATI" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="536" data-original-width="578" height="371" src="https://blogger.googleusercontent.com/img/a/AVvXsEhW7lcGzMtZ9v4cqHACnWtKHX_JU8vMf8dS4abiio_-PCxuVdSm5J9kAkoSQja4JSgkRLjNhEdtfvYqfBz7uY_qFdejXAHdTNDi4b4Ve7aD9_yudLxNz78UA-PFd3-S7DEvezWRB1MkZDglkMcQc-3aOZ9Nr1Oz7IKod3dUj_tszoFLwvQj3gxAmy_zAATI=w400-h371" width="400" /></a></div><br />Birds...</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhOU6AfcgGGEUGfH9__1bSdEqNrA6nMR2DI3MFcTCk2SwWtOsjgT0-UJ9Ws9XOtf76qKU5lECdVwR9RVI7Sbu5RB3FlxTaRI685mvGzXJQL-5RVIP8z62FMxF5zG3-Sb2i8r7mE8Rb_nZENY_CmN28kt_s2Dw0SJrjPcOyHrBPZhHnR-Bo4yXApTlLlvjKN" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEhOU6AfcgGGEUGfH9__1bSdEqNrA6nMR2DI3MFcTCk2SwWtOsjgT0-UJ9Ws9XOtf76qKU5lECdVwR9RVI7Sbu5RB3FlxTaRI685mvGzXJQL-5RVIP8z62FMxF5zG3-Sb2i8r7mE8Rb_nZENY_CmN28kt_s2Dw0SJrjPcOyHrBPZhHnR-Bo4yXApTlLlvjKN=w400-h225" width="400" /></a></div><br /><div class="separator" style="clear: both; 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text-align: center;"><br /></div></div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-10136518262782194912023-12-13T10:22:00.000-08:002023-12-13T16:00:05.864-08:00Murmurations Dawn 13 December 2023<div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibnmbYa2Deyzecax6j4u-RbGFMMoKmcB5W5NVjR9N22qgyU_ol0LCyqS8Xym2GuqvQOICZwBkMnY2Pan_JYNLpU2pedAoAqCOueM1T09robNIyEmNhw4FKqvEqsgA1pBbVypETtD7SQ70IC6xEzgZRxhR0clyj8y_EOS5s_noymyGf8RvNsRPGxnZcK51J/s1300/Screenshot%20from%202023-12-13%2011-18-06.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibnmbYa2Deyzecax6j4u-RbGFMMoKmcB5W5NVjR9N22qgyU_ol0LCyqS8Xym2GuqvQOICZwBkMnY2Pan_JYNLpU2pedAoAqCOueM1T09robNIyEmNhw4FKqvEqsgA1pBbVypETtD7SQ70IC6xEzgZRxhR0clyj8y_EOS5s_noymyGf8RvNsRPGxnZcK51J/w400-h225/Screenshot%20from%202023-12-13%2011-18-06.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn8RMNowfQWV6xdh5hxivfE-Jhg1GaYvyhh6Gehs9N0Ufc_yXn4TZmIKNe3dpW_b9GiB3vxR_jErBZVXvSQR5zFc_1iEDFSuECkMfHK7nBc_afxL3JvS3DlnERcyByIQYTrn7mEGWUUUKUO68YOd7l3-Zk5aI6dCsRamz2StQLmYAXB4YL98c9P6EPEt86/s1300/Screenshot%20from%202023-12-13%2011-26-49.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn8RMNowfQWV6xdh5hxivfE-Jhg1GaYvyhh6Gehs9N0Ufc_yXn4TZmIKNe3dpW_b9GiB3vxR_jErBZVXvSQR5zFc_1iEDFSuECkMfHK7nBc_afxL3JvS3DlnERcyByIQYTrn7mEGWUUUKUO68YOd7l3-Zk5aI6dCsRamz2StQLmYAXB4YL98c9P6EPEt86/w400-h225/Screenshot%20from%202023-12-13%2011-26-49.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOLMNjhGpBLCAxh0PjlVEevd-cU99j6Kppg85mQiarzZr3GUeH8rJBMV0SBQ3ahPUvu20tf3PP2fuoMvLFy82m7Z7YeHq7qHOVqu4EoNTfds7AunlbG8FwKYdhiieo3Iaq6nszfStR0BctWYhkqcwTtGAj0sld1wuq81ixQsuwcMkQCAXdzKya3V6whQNV/s1300/Screenshot%20from%202023-12-13%2011-20-28.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOLMNjhGpBLCAxh0PjlVEevd-cU99j6Kppg85mQiarzZr3GUeH8rJBMV0SBQ3ahPUvu20tf3PP2fuoMvLFy82m7Z7YeHq7qHOVqu4EoNTfds7AunlbG8FwKYdhiieo3Iaq6nszfStR0BctWYhkqcwTtGAj0sld1wuq81ixQsuwcMkQCAXdzKya3V6whQNV/w400-h225/Screenshot%20from%202023-12-13%2011-20-28.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjj0QjzZuJDTVVBqY0nAQv2JWetGYPKq4X2qw_xc2rjg2J1Gb78YnD4nwUkRUtZeRLnb1JhErG2_2S5KVvoebxgtqq6W7u0j3foH2tDSpU2C7Yo0o2l8eVHe8nZ6xiYfVDim72EtP7zhsxBNd693rrnizBmqoTYrbCynpA1NzoJgvwe0mCk-g89Kvk6zZCH" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEjj0QjzZuJDTVVBqY0nAQv2JWetGYPKq4X2qw_xc2rjg2J1Gb78YnD4nwUkRUtZeRLnb1JhErG2_2S5KVvoebxgtqq6W7u0j3foH2tDSpU2C7Yo0o2l8eVHe8nZ6xiYfVDim72EtP7zhsxBNd693rrnizBmqoTYrbCynpA1NzoJgvwe0mCk-g89Kvk6zZCH=w400-h225" width="400" /></a></div><br /><br /></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-53924333053683067872023-12-13T05:00:00.000-08:002023-12-13T10:03:14.513-08:00Counting Objects continuedFrom my previous chat session, I asked chatGPT to generate five prompts that would provide it with adequate instructions to do all of this in ONE step. Here's what it generated:<div><br /></div><div><div><i>"Please apply a 10x10 pixel median filter to an uploaded grayscale image, then perform double thresholding with a minimum threshold of 0.1 and a maximum threshold of 0.5. Count the number of objects, label each object with a unique, incrementing number directly on the image, and display the histogram of pixel intensities with a logarithmic y-axis scale. Show the median-filtered image, the thresholded and labeled image, and the histogram in your response."</i></div><div><i><br /></i></div><div><i>"Load an uploaded image, convert it to grayscale if it isn't already, and apply a median filter using a 10x10 pixel window. After filtering, use double thresholding with thresholds set at 0.1 and 0.5. Count the distinct objects in the thresholded image and label them sequentially on the image itself. Additionally, generate and display a histogram of the median-filtered image with a logarithmic scale on the y-axis. Provide the filtered image, the labeled thresholded image, and the histogram in your final output."</i></div><div><i><br /></i></div><div><i>"I need you to process an uploaded image as follows: first, apply a median filter with a 10x10 kernel, then conduct a double thresholding operation with thresholds of 0.1 and 0.5. Count and label each detected object in the image, using incremental numbers for labeling. Also, produce a histogram of the filtered image's pixel intensities on a logarithmic y-axis. Present the original median-filtered image, the labeled image after thresholding, and the histogram in your response."</i></div><div><i><br /></i></div><div><i>"Start by applying a 10x10 median filter to an uploaded grayscale image. Then perform thresholding with a minimum limit of 0.1 and a maximum of 0.5. Identify and label each object in the image with increasing numbers. Also, create a histogram showing the distribution of pixel intensities from the median-filtered image on a log scale. Display the processed median-filtered image, the final labeled image after thresholding, and the histogram."</i></div><div><i><br /></i></div><div><i>"For an uploaded grayscale image, first implement a median filtering with a 10x10 window. Next, apply a double thresholding technique with thresholds set at 0.1 (minimum) and 0.5 (maximum). Count the objects in the thresholded image, labeling each with a sequential number. Generate a histogram of the pixel intensities post-median filtering using a logarithmic scale for the frequency. Please provide the original median-filtered image, the thresholded and labeled image, and the histogram in your results."</i></div></div><div><br /></div><div>I used the first prompt, and here is the result:</div><div><br /></div><div>uploaded image:</div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhlccTCKqTnhLLc6MKX75ExDD-SiXuUNFWzoRgeSZeGaJ1FlL-v3CqbdFu0NBe-nWuJQq0gytICc6uxDSDfWQ4B1IF7Wyqin6UV6np88WaXQiFSVMPhu8rBjU0X33PqPQKw_yp6Ik6Hyve-lMJRR2_UHSIctPEZ_qF8uqws3fvENKjaqyBYY6LOfGQCOXiQ" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1224" data-original-width="1632" height="300" src="https://blogger.googleusercontent.com/img/a/AVvXsEhlccTCKqTnhLLc6MKX75ExDD-SiXuUNFWzoRgeSZeGaJ1FlL-v3CqbdFu0NBe-nWuJQq0gytICc6uxDSDfWQ4B1IF7Wyqin6UV6np88WaXQiFSVMPhu8rBjU0X33PqPQKw_yp6Ik6Hyve-lMJRR2_UHSIctPEZ_qF8uqws3fvENKjaqyBYY6LOfGQCOXiQ=w400-h300" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div style="text-align: justify;">Me: Please apply a 10x10 pixel median filter to an uploaded grayscale image, then perform double thresholding with a minimum threshold of 0.1 and a maximum threshold of 0.5. Count the number of objects, label each object with a unique, incrementing number directly on the image, and display the histogram of pixel intensities with a logarithmic y-axis scale. Show the median-filtered image, the thresholded and labeled image, and the histogram in your response.</div><div style="text-align: justify;"><br /></div></div><div style="text-align: justify;">chatGPT:</div><div style="text-align: justify;"><br /></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">import cv2</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">import numpy as np</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">import matplotlib.pyplot as plt</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Load the image</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">image_path = '/mnt/data/pennies.png'</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">original_image = cv2.imread(image_path, cv2.IMREAD_GRAYSCALE)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Apply a 10x10 median filter</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">median_filtered = cv2.medianBlur(original_image, 10)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Perform double thresholding</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Normalizing the pixel intensities to be between 0 and 1 for thresholding</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">normalized_img = median_filtered / 255.0</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">thresholded = np.zeros_like(normalized_img)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">thresholded[(normalized_img >= 0.1) & (normalized_img <= 0.5)] = 1</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Find and label the objects</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">num_labels, labels_im = cv2.connectedComponents(thresholded.astype(np.uint8))</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Create an image that will be used to display the labels</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">label_hue = np.uint8(179*labels_im/np.max(labels_im))</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">blank_ch = 255*np.ones_like(label_hue)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">labeled_img = cv2.merge([label_hue, blank_ch, blank_ch])</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">labeled_img = cv2.cvtColor(labeled_img, cv2.COLOR_HSV2BGR)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">labeled_img[label_hue==0] = 0</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Draw the labels on the image</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">for i in range(1, num_labels):</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"> cv2.putText(labeled_img, str(i), tuple(np.mean(np.argwhere(labels_im == i), axis=0).astype(int)),</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"> cv2.FONT_HERSHEY_SIMPLEX, 0.5, (255,255,255), 1)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Create a histogram of pixel intensities with a logarithmic y-axis scale</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">histogram, bin_edges = np.histogram(original_image, bins=256, range=(0, 256))</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Plotting the histogram</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.figure()</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.plot(bin_edges[0:-1], histogram, color='black')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.yscale('log')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.title('Histogram of Pixel Intensities (Log Scale)')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.xlabel('Pixel Intensity')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.ylabel('Frequency (Log Scale)')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Save the images and histogram</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">median_filtered_path = '/mnt/data/median_filtered.png'</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">cv2.imwrite(median_filtered_path, median_filtered)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">labeled_img_path = '/mnt/data/labeled_img.png'</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">cv2.imwrite(labeled_img_path, labeled_img)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">histogram_path = '/mnt/data/histogram.png'</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.savefig(histogram_path)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.close()</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Return the paths to the saved images and histogram</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">median_filtered_path, labeled_img_path, histogram_path, num_labels - 1</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><div><span style="font-family: courier; font-size: x-small;">STDOUT/STDERR</span></div></div></div></blockquote><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px;"><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">Matplotlib is building the font cache; this may take a moment.</span></div></div></blockquote><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><span face="Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"" style="color: #374151; font-size: 16px; text-align: start; white-space: pre-wrap;">It looks like there was an issue applying a median filter with a kernel size of 10, which must be an odd number. I will correct this and use a 9x9 kernel instead, which is the nearest odd kernel size to 10. Let's proceed with the corrected filter size. </span></div><div style="text-align: justify;"><br /></div><blockquote style="border: none; margin: 0px 0px 0px 40px; padding: 0px; text-align: left;"><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">import cv2</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">import numpy as np</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">import matplotlib.pyplot as plt</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Load the image</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">image_path = '/mnt/data/pennies.png'</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">original_image = cv2.imread(image_path, cv2.IMREAD_GRAYSCALE)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Apply a 10x10 median filter</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">median_filtered = cv2.medianBlur(original_image, 10)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Perform double thresholding</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Normalizing the pixel intensities to be between 0 and 1 for thresholding</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">normalized_img = median_filtered / 255.0</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">thresholded = np.zeros_like(normalized_img)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">thresholded[(normalized_img >= 0.1) & (normalized_img <= 0.5)] = 1</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Find and label the objects</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">num_labels, labels_im = cv2.connectedComponents(thresholded.astype(np.uint8))</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Create an image that will be used to display the labels</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">label_hue = np.uint8(179*labels_im/np.max(labels_im))</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">blank_ch = 255*np.ones_like(label_hue)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">labeled_img = cv2.merge([label_hue, blank_ch, blank_ch])</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">labeled_img = cv2.cvtColor(labeled_img, cv2.COLOR_HSV2BGR)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">labeled_img[label_hue==0] = 0</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Draw the labels on the image</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">for i in range(1, num_labels):</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"> cv2.putText(labeled_img, str(i), tuple(np.mean(np.argwhere(labels_im == i), axis=0).astype(int)),</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"> cv2.FONT_HERSHEY_SIMPLEX, 0.5, (255,255,255), 1)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Create a histogram of pixel intensities with a logarithmic y-axis scale</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">histogram, bin_edges = np.histogram(original_image, bins=256, range=(0, 256))</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Plotting the histogram</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.figure()</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.plot(bin_edges[0:-1], histogram, color='black')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.yscale('log')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.title('Histogram of Pixel Intensities (Log Scale)')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.xlabel('Pixel Intensity')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.ylabel('Frequency (Log Scale)')</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Save the images and histogram</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">median_filtered_path = '/mnt/data/median_filtered.png'</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">cv2.imwrite(median_filtered_path, median_filtered)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">labeled_img_path = '/mnt/data/labeled_img.png'</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">cv2.imwrite(labeled_img_path, labeled_img)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">histogram_path = '/mnt/data/histogram.png'</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.savefig(histogram_path)</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">plt.close()</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"><br /></span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;"># Return the paths to the saved images and histogram</span></div></div><div style="text-align: justify;"><div><span style="font-family: courier; font-size: x-small;">median_filtered_path, labeled_img_path, histogram_path, num_labels - 1</span></div></div></blockquote><div style="text-align: justify;"><div><br /></div></div><div style="text-align: justify;"><div><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: #374151; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 0px 0px 1.25em; text-align: start; white-space: pre-wrap;">The median filtering, thresholding, object labeling, and histogram generation have been completed successfully. Here are the results:</p><ul style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: #374151; display: flex; flex-direction: column; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 1.25em 0px; padding: 0px; text-align: start; white-space: pre-wrap;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: block; margin: 0px; min-height: 28px; padding-left: 0.375em;"><span color="var(--tw-prose-bold)" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600; margin-bottom: 1.25em; margin-top: 1.25em;">Median-Filtered Image</span>: This image has been filtered using a 9x9 median filter.</li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: block; margin: 0px; min-height: 28px; padding-left: 0.375em;"><span color="var(--tw-prose-bold)" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600; margin-bottom: 1.25em; margin-top: 1.25em;">Thresholded and Labeled Image</span>: The objects have been identified and labeled with unique, incrementing numbers.</li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: block; margin: 0px; min-height: 28px; padding-left: 0.375em;"><span color="var(--tw-prose-bold)" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600; margin-bottom: 1.25em; margin-top: 1.25em;">Histogram of Pixel Intensities</span>: The histogram displays the frequency of pixel intensities on a logarithmic scale.</li></ul><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: #374151; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; text-align: start; white-space: pre-wrap;">There are a total of 13 objects identified in the image.</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: #374151; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; margin: 1.25em 0px; text-align: start; white-space: pre-wrap;">You can view and download the images and histogram using the following links:</p><ul style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: #374151; display: flex; flex-direction: column; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; list-style-image: initial; list-style-position: initial; margin: 1.25em 0px; padding: 0px; text-align: start; white-space: pre-wrap;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: block; margin: 0px; min-height: 28px; padding-left: 0.375em;"><span data-state="closed" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin-bottom: 1.25em; margin-top: 1.25em;"><a class="cursor-pointer" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(41,100,170,var(--tw-text-opacity)); cursor: pointer;" target="_new">Median-Filtered Image</a></span></li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: block; margin: 0px; min-height: 28px; padding-left: 0.375em;"><span data-state="closed" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin-bottom: 1.25em; margin-top: 1.25em;"><a class="cursor-pointer" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(41,100,170,var(--tw-text-opacity)); cursor: pointer;" target="_new">Thresholded and Labeled Image</a></span></li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: block; margin: 0px; min-height: 28px; padding-left: 0.375em;"><span data-state="closed" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin-bottom: 1.25em; margin-top: 1.25em;"><a class="cursor-pointer" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(41,100,170,var(--tw-text-opacity)); cursor: pointer;" target="_new">Histogram</a></span></li></ul></div><div><div class="min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&]:mt-5 overflow-x-auto" data-message-author-role="assistant" data-message-id="b77dbb93-e220-484d-9572-8f7d22dc2a7e" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-start; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: #0f0f0f; display: flex; flex-direction: column; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; gap: 0.75rem; margin-top: 1.25rem; min-height: 20px; overflow-wrap: break-word; overflow-x: auto; text-align: start; white-space: pre-wrap;"><div class="markdown prose w-full break-words dark:prose-invert light" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-prose-body: #374151; --tw-prose-bold: #111827; --tw-prose-bullets: #d1d5db; --tw-prose-captions: #6b7280; --tw-prose-code: #111827; --tw-prose-counters: #6b7280; --tw-prose-headings: #111827; --tw-prose-hr: #e5e7eb; --tw-prose-invert-body: #d1d5db; --tw-prose-invert-bold: #fff; --tw-prose-invert-bullets: #4b5563; --tw-prose-invert-captions: #9ca3af; --tw-prose-invert-code: #fff; --tw-prose-invert-counters: #9ca3af; --tw-prose-invert-headings: #fff; --tw-prose-invert-hr: #374151; --tw-prose-invert-lead: #9ca3af; --tw-prose-invert-links: #fff; --tw-prose-invert-pre-bg: rgba(0,0,0,0.5); --tw-prose-invert-pre-code: #d1d5db; --tw-prose-invert-quote-borders: #374151; --tw-prose-invert-quotes: #f3f4f6; --tw-prose-invert-td-borders: #374151; --tw-prose-invert-th-borders: #4b5563; --tw-prose-lead: #4b5563; --tw-prose-links: #111827; --tw-prose-pre-bg: #1f2937; --tw-prose-pre-code: #e5e7eb; --tw-prose-quote-borders: #e5e7eb; --tw-prose-quotes: #111827; --tw-prose-td-borders: #e5e7eb; --tw-prose-th-borders: #d1d5db; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: var(--tw-prose-body); font-size: 1rem; line-height: 1.75; max-width: none; overflow-wrap: break-word; width: 613px;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px 0px;">Now I will display the images and histogram inline for your convenience. <span data-state="closed" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><button style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="-mt-0.5 ml-0.5 inline-block text-link-base hover:text-link-hover" fill="none" height="15" viewbox="0 0 19 15" width="19" xmlns="http://www.w3.org/2000/svg"><path d="M4.42 0.75H2.8625H2.75C1.64543 0.75 0.75 1.64543 0.75 2.75V11.65C0.75 12.7546 1.64543 13.65 2.75 13.65H2.8625C2.8625 13.65 2.8625 13.65 2.8625 13.65C2.8625 13.65 4.00751 13.65 4.42 13.65M13.98 13.65H15.5375H15.65C16.7546 13.65 17.65 12.7546 17.65 11.65V2.75C17.65 1.64543 16.7546 0.75 15.65 0.75H15.5375H13.98" stroke-linecap="round" stroke-linejoin="round" stroke-width="1.5" stroke="currentColor"></path><path d="M5.55283 4.21963C5.25993 3.92674 4.78506 3.92674 4.49217 4.21963C4.19927 4.51252 4.19927 4.9874 4.49217 5.28029L6.36184 7.14996L4.49217 9.01963C4.19927 9.31252 4.19927 9.7874 4.49217 10.0803C4.78506 10.3732 5.25993 10.3732 5.55283 10.0803L7.95283 7.68029C8.24572 7.3874 8.24572 6.91252 7.95283 6.61963L5.55283 4.21963Z" fill="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="0.2" stroke="currentColor"></path><path d="M9.77753 8.75003C9.3357 8.75003 8.97753 9.10821 8.97753 9.55003C8.97753 9.99186 9.3357 10.35 9.77753 10.35H13.2775C13.7194 10.35 14.0775 9.99186 14.0775 9.55003C14.0775 9.10821 13.7194 8.75003 13.2775 8.75003H9.77753Z" fill="currentColor" stroke-width="0.1" stroke="currentColor"></path></svg></button></span></p></div></div><div class="min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&]:mt-5 overflow-x-auto" data-message-author-role="assistant" data-message-id="40265c9f-f495-4116-b9c2-6e802648ace7" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-start; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: #0f0f0f; display: flex; flex-direction: column; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px; gap: 0.75rem; min-height: 20px; overflow-wrap: break-word; overflow-x: auto; text-align: start; white-space: pre-wrap;"><div class="markdown prose w-full break-words dark:prose-invert light" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-prose-body: #374151; --tw-prose-bold: #111827; --tw-prose-bullets: #d1d5db; --tw-prose-captions: #6b7280; --tw-prose-code: #111827; --tw-prose-counters: #6b7280; --tw-prose-headings: #111827; --tw-prose-hr: #e5e7eb; --tw-prose-invert-body: #d1d5db; --tw-prose-invert-bold: #fff; --tw-prose-invert-bullets: #4b5563; --tw-prose-invert-captions: #9ca3af; --tw-prose-invert-code: #fff; --tw-prose-invert-counters: #9ca3af; --tw-prose-invert-headings: #fff; --tw-prose-invert-hr: #374151; --tw-prose-invert-lead: #9ca3af; --tw-prose-invert-links: #fff; --tw-prose-invert-pre-bg: rgba(0,0,0,0.5); --tw-prose-invert-pre-code: #d1d5db; --tw-prose-invert-quote-borders: #374151; --tw-prose-invert-quotes: #f3f4f6; --tw-prose-invert-td-borders: #374151; --tw-prose-invert-th-borders: #4b5563; --tw-prose-lead: #4b5563; --tw-prose-links: #111827; --tw-prose-pre-bg: #1f2937; --tw-prose-pre-code: #e5e7eb; --tw-prose-quote-borders: #e5e7eb; --tw-prose-quotes: #111827; --tw-prose-td-borders: #e5e7eb; --tw-prose-th-borders: #d1d5db; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: var(--tw-prose-body); font-size: 1rem; line-height: 1.75; max-width: none; overflow-wrap: break-word; width: 613px;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px 0px 1.25em;">above are the processed images and histogram:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px 0px 1.25em;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjj-66-obUeNDgpmqSNtgj83w0GUy6Hc0GACbNVyvGfUdNb51gIsC8zLzsRl_FJBWOKOuczdIdMDHg31ITqycsQc3rVZnhTaaMJAtvVZy9-DP2fava6ZmOqkuDU4pYW3g4C7UQ4x7HNjd4AX6B4XERfEys1nGFR7oMm-BNBrapSB8PzK98J-gXMdLcyBFka" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1224" data-original-width="1632" height="150" src="https://blogger.googleusercontent.com/img/a/AVvXsEjj-66-obUeNDgpmqSNtgj83w0GUy6Hc0GACbNVyvGfUdNb51gIsC8zLzsRl_FJBWOKOuczdIdMDHg31ITqycsQc3rVZnhTaaMJAtvVZy9-DP2fava6ZmOqkuDU4pYW3g4C7UQ4x7HNjd4AX6B4XERfEys1nGFR7oMm-BNBrapSB8PzK98J-gXMdLcyBFka=w200-h150" width="200" /></a><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhz1C3z0_rJxeNsMkfQkM3bE94mciQU70n-XAL6WqpWNzo1jwrw63VGBlIS3oFtJ0BOkIX4t78CqjEnl2ZMlLGgJ63OiuynYJ4lb59GeeEg1aNK10JYRrJYEbs1FZGTBenrMh_71pyKKeVCpiDdgZXIOogamhzQgaksxrIcSZpMuJyVYuAmaL7-ff9uabQJ" style="font-size: 1rem; margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1224" data-original-width="1632" height="150" src="https://blogger.googleusercontent.com/img/a/AVvXsEhz1C3z0_rJxeNsMkfQkM3bE94mciQU70n-XAL6WqpWNzo1jwrw63VGBlIS3oFtJ0BOkIX4t78CqjEnl2ZMlLGgJ63OiuynYJ4lb59GeeEg1aNK10JYRrJYEbs1FZGTBenrMh_71pyKKeVCpiDdgZXIOogamhzQgaksxrIcSZpMuJyVYuAmaL7-ff9uabQJ=w200-h150" width="200" /></a><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhd38iYYAFgZJurSq3u4xbtiUfuJTZpmnff4kfemShEBXsn-18bfRS6DBS8TEjYxycKVLGKKP8OD3kCQRHHu6a7xOO3kSL0teE-U5HunqQvp1CxL2pU4q8aX-lclr37tHmpJ0AVZ7VNss10YCaUWKxUnpZmhiFcBMHHZgfU2suZ9u7khTx2YXJrRVHdmLHe" style="font-size: 1rem; margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1200" data-original-width="2000" height="120" src="https://blogger.googleusercontent.com/img/a/AVvXsEhd38iYYAFgZJurSq3u4xbtiUfuJTZpmnff4kfemShEBXsn-18bfRS6DBS8TEjYxycKVLGKKP8OD3kCQRHHu6a7xOO3kSL0teE-U5HunqQvp1CxL2pU4q8aX-lclr37tHmpJ0AVZ7VNss10YCaUWKxUnpZmhiFcBMHHZgfU2suZ9u7khTx2YXJrRVHdmLHe=w200-h120" width="200" /></a></div><br /><br /><p></p><ol style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-reset: list-number 0; display: flex; flex-direction: column; list-style: none; margin: 1.25em 0px 0px; padding: 0px;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;">On the left is the <span color="var(--tw-prose-bold)" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600; margin-bottom: 1.25em; margin-top: 1.25em;">Median-Filtered Image</span>, which has been smoothed using a 9x9 median filter.</li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;">In the middle is the <span color="var(--tw-prose-bold)" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600; margin-bottom: 1.25em; margin-top: 1.25em;">Thresholded and Labeled Image</span>, where each object detected within the specified thresholds is labeled with a unique number.</li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;">On the right is the <span color="var(--tw-prose-bold)" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600; margin-top: 1.25em;">Histogram of Pixel Intensities</span> with a logarithmic y-axis scale, showing the distribution of pixel values within the original image. </li></ol></div></div></div></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">---------------------</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Ok, so as you can see, it correctly counts the number of objects. There are some notable problems, which I'll focus on in my next post.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">For now I'm very pleased with the results. I want to try the four other prompts and different input images with objects to count.</div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-85287120682556444402023-12-12T14:17:00.000-08:002023-12-12T14:17:18.270-08:00Sandhill Crane Flight Long ExposuresThis is 38 seconds of video, with every 5th image captured and stacked:<div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi04i-nuVuXDCms9y8_WswwcMs7AmLhbhgdLScgyrqvE8gUzoNzUrfv5A5lfyJyVtJY0hI0eS9v_reuEjnP-beBcSf0VdoeB3NIMsnJqFqBj-o0m7tIgW8hDQAAXJffVUb5cYRTDYdvOQEI1a_UdkvJxN9PQqeBv1tGG0dayLtCCskjHUA2f6VBOgmf2jNs" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEi04i-nuVuXDCms9y8_WswwcMs7AmLhbhgdLScgyrqvE8gUzoNzUrfv5A5lfyJyVtJY0hI0eS9v_reuEjnP-beBcSf0VdoeB3NIMsnJqFqBj-o0m7tIgW8hDQAAXJffVUb5cYRTDYdvOQEI1a_UdkvJxN9PQqeBv1tGG0dayLtCCskjHUA2f6VBOgmf2jNs=w400-h225" width="400" /></a></div><br /><br /></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-51134585419240604702023-12-11T18:40:00.000-08:002023-12-11T18:40:02.674-08:00Sandhill Crane Murmurations<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgsXf8Qfc-NYK7XVOaxi6Jn-IKJlBFWfmBnZv2x6b1Dktvds9xVEQ2Jdzk-0qEVZl2HWqoIqI3Vbdo7qefjPoW9zFF6CrV8mS1doYxvvIqpW3yZpOhaWqGk4CcmYvRexIndUYr1blpMU7xKAzMi8bX01MKZoEN30KGbJyUiOWGCDUzwhF72D4jGTOMe4wK0" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="800" data-original-width="450" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEgsXf8Qfc-NYK7XVOaxi6Jn-IKJlBFWfmBnZv2x6b1Dktvds9xVEQ2Jdzk-0qEVZl2HWqoIqI3Vbdo7qefjPoW9zFF6CrV8mS1doYxvvIqpW3yZpOhaWqGk4CcmYvRexIndUYr1blpMU7xKAzMi8bX01MKZoEN30KGbJyUiOWGCDUzwhF72D4jGTOMe4wK0=w225-h400" width="225" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhSjWD77TeTEwpc72PbgQRgqmhC9FnavgcGupdzElssdIqR3C9EqcOQHdQVvygN-6OB01U_049RufeMI3_8doZR0Qn_JT_-cg4Ej1C4TF6Uoj4MgdQHugqsc9mbOx8ZyriSfatQvyUDSXNOfJC8IqRUpORwq3VRsUNTQOXUetPkqmF6GDOu5ah2M0w2Zeo-" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEhSjWD77TeTEwpc72PbgQRgqmhC9FnavgcGupdzElssdIqR3C9EqcOQHdQVvygN-6OB01U_049RufeMI3_8doZR0Qn_JT_-cg4Ej1C4TF6Uoj4MgdQHugqsc9mbOx8ZyriSfatQvyUDSXNOfJC8IqRUpORwq3VRsUNTQOXUetPkqmF6GDOu5ah2M0w2Zeo-=w400-h225" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgTKMsQc6AZvtugHT4kfcthLEqHMg6XnPxb4ien3RwVw2-kCj76-qYos7fGjccs_Njh3Uf2lvUQVaDurXVqcys0mqu6bEa3H5b64ePhSAcxNnSNZD6MFORdFjiTn4NVgfLIywn76PcobxYT5VEUXQQ2QWzpCphZv4FgN0LTwl88AD2DCcF60UA-SkCXwwte" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="800" data-original-width="450" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEgTKMsQc6AZvtugHT4kfcthLEqHMg6XnPxb4ien3RwVw2-kCj76-qYos7fGjccs_Njh3Uf2lvUQVaDurXVqcys0mqu6bEa3H5b64ePhSAcxNnSNZD6MFORdFjiTn4NVgfLIywn76PcobxYT5VEUXQQ2QWzpCphZv4FgN0LTwl88AD2DCcF60UA-SkCXwwte=w225-h400" width="225" /></a></div><br /><br />Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-37545328545330287102023-12-11T14:50:00.000-08:002023-12-11T15:10:03.795-08:00Counting ObjectsIf an assistant can't count with the accuracy and precision I require, then it's useless to me as a researcher. This is true whether it's a human or an AI. So I consider counting objects in an image as a benchmark test.<div><br /></div><div>For AI, sofar it's failed. chatGPT inaccuractly counts objects in an image that any three-year-old human can count with ease.</div><div><br /></div><div>Up until now, I've been using this input image as a test, and I've asked chatGPT to count the number of objects in this image:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjPKCQNIXzC_O184Ul4UCteNG2LLnQeiGUcCRsnFPAHL3eR7_XbzDHsQqGRBmFe6H2Y2YXkReCv5KUfUCgpSs16P6LdzbRCSAD5vCiCYJH_1AdJvltYeDz4ecVPXYUgDRnCB_anq4P3Mr0fIOyiWPRRIC8BMTnHvU2_TuqpESXBaXSWvqQL1FdGeL82SHUF" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1536" data-original-width="1152" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEjPKCQNIXzC_O184Ul4UCteNG2LLnQeiGUcCRsnFPAHL3eR7_XbzDHsQqGRBmFe6H2Y2YXkReCv5KUfUCgpSs16P6LdzbRCSAD5vCiCYJH_1AdJvltYeDz4ecVPXYUgDRnCB_anq4P3Mr0fIOyiWPRRIC8BMTnHvU2_TuqpESXBaXSWvqQL1FdGeL82SHUF=w300-h400" width="300" /></a></div><br />The main problems with this image are:</div><div><br /></div><div>1. there are a number of ghost-like reflections of nuts that are produced by a pretty terrible optical system. The explanation for this is that there's a crack going through the center of the camera's field of view, producing these ghost images. So how many nuts are there? Hard to tell and not exactly fair.</div><div><br /></div><div>2. complex background - not only is the background a wood grain, the lighting isn't uniform either.</div><div><br /></div><div>So how many objects are there? Haha it really does depend on attention.</div><div><br /></div><div>So I started with something else instead. In this case, thirteen pennies on a fairly uniform background.</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEizOdrkMDEMYUTTLLUX7Gph2IHNRr4LtI3dEEF2frvMnJqea6s3YF9II_gRpzdXBDABYyDNxyixJ5fjeKdmC53V3TUXR06daTwzKJg1wym5MlZ4uakL_Me9jQVPnp0VMXkEIIymZKv1qEV-9rH3EoBSQ11YE8s4gqnHVIDZkS18CXrcMnb6-P7fT-4JszAJ" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="514" data-original-width="662" height="311" src="https://blogger.googleusercontent.com/img/a/AVvXsEizOdrkMDEMYUTTLLUX7Gph2IHNRr4LtI3dEEF2frvMnJqea6s3YF9II_gRpzdXBDABYyDNxyixJ5fjeKdmC53V3TUXR06daTwzKJg1wym5MlZ4uakL_Me9jQVPnp0VMXkEIIymZKv1qEV-9rH3EoBSQ11YE8s4gqnHVIDZkS18CXrcMnb6-P7fT-4JszAJ=w400-h311" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">I've noticed that 'errors' seem to happen pretty often. I'm pretty sure it's just overload on the server. The fact that it's checking it's work is nice but I can image that might also be annoying.</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Anyhow it's first attempt at thresholding to segment this image is pretty good:</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhxswkKzhegNGP71F9VCv4RxwYk6pth1g7uBdtvqRj9ZYqg68-x64vawQcx9H_FENdJkWMT-yH3IQaG_RCklggSjUxnnhuIRpGQRKWsj09xmEU2ZWxycl_yJcApqvyEtiJtoihoXKhFoHfa3mt5MrVgpg5GM9tGKkvtIWScj2xAwUHe-fTgwa7Y5oz8g797" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="565" data-original-width="670" height="337" src="https://blogger.googleusercontent.com/img/a/AVvXsEhxswkKzhegNGP71F9VCv4RxwYk6pth1g7uBdtvqRj9ZYqg68-x64vawQcx9H_FENdJkWMT-yH3IQaG_RCklggSjUxnnhuIRpGQRKWsj09xmEU2ZWxycl_yJcApqvyEtiJtoihoXKhFoHfa3mt5MrVgpg5GM9tGKkvtIWScj2xAwUHe-fTgwa7Y5oz8g797=w400-h337" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Here I ask it to produce a histogram and it does...</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj8DrZOC1nQS5SD9FwgcpobvlXIVH2xWeLCXmyuJjVfBFV_xUfZdfY1CKie4TOQXlS0eESccBzX0qCLVt-k6OTjD4sFnrkAFudxmmK4_FVAITp0ScMAd_puRsFRWbV-WlXQ08mc7yYew9nS3ohNVt_N6-spOl5PTJ95U6OdEilON0ZJmkL7ffpNyvct2sdS" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="556" data-original-width="672" height="331" src="https://blogger.googleusercontent.com/img/a/AVvXsEj8DrZOC1nQS5SD9FwgcpobvlXIVH2xWeLCXmyuJjVfBFV_xUfZdfY1CKie4TOQXlS0eESccBzX0qCLVt-k6OTjD4sFnrkAFudxmmK4_FVAITp0ScMAd_puRsFRWbV-WlXQ08mc7yYew9nS3ohNVt_N6-spOl5PTJ95U6OdEilON0ZJmkL7ffpNyvct2sdS=w400-h331" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Easily able to change the thresholding values...</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhZNDFpBtKCD0mr6g5it62_HZo43Oia108UWJyiNg2H4JfXJmoFBT2YY0xpUOW1TheHxIo64GxRxe30I-IyRAVcrrEevUslKZEwDmH-_mCRG6TaqGw8UJGoWYxIJIbu4HV-DxkPvVX_CTb7o7JS2Pw7030iqncrRvmUZG6IJnzJAPs9ceOT0P2BQunnFyFF" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="602" data-original-width="662" height="364" src="https://blogger.googleusercontent.com/img/a/AVvXsEhZNDFpBtKCD0mr6g5it62_HZo43Oia108UWJyiNg2H4JfXJmoFBT2YY0xpUOW1TheHxIo64GxRxe30I-IyRAVcrrEevUslKZEwDmH-_mCRG6TaqGw8UJGoWYxIJIbu4HV-DxkPvVX_CTb7o7JS2Pw7030iqncrRvmUZG6IJnzJAPs9ceOT0P2BQunnFyFF=w400-h364" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjs7uy0KvUF-LNmzVX5iCJPcxF6ijIUTaFIL9UEoIsbf-KW73vG1LZTQGIPFoU_C3BWdnmmqw4LWVAwtfBpM7PGEXp-9byuONK9vWZqoYZUVfOxlTGBDCyyMlgoHBcgXr0wlkRhUyJyjVWOdW-NWu04T7WPAgveHot2CvoZJsrPY3fCNYMpBg0t4rkRoypq" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="603" data-original-width="664" height="364" src="https://blogger.googleusercontent.com/img/a/AVvXsEjs7uy0KvUF-LNmzVX5iCJPcxF6ijIUTaFIL9UEoIsbf-KW73vG1LZTQGIPFoU_C3BWdnmmqw4LWVAwtfBpM7PGEXp-9byuONK9vWZqoYZUVfOxlTGBDCyyMlgoHBcgXr0wlkRhUyJyjVWOdW-NWu04T7WPAgveHot2CvoZJsrPY3fCNYMpBg0t4rkRoypq=w400-h364" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">I'm even able to tell it to change the scale from linear to logarithmic and poof it does it!</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhRpa47xlPxWsNqUte_9M-2bIkYY4mfBsRN-GY4jm3fsfL5v9Ev8mjfFQ5kTXXD0Ji54srkZzhtCyMWi680U8bd5-6pg4KMuOUw8AJYCkBdDdmzrkCov9if74NdqK0pa_fxCxHfmJMqd_S4NeREHfKyxN8SVCZOfKq1R_dqFs3KQyRgXtDPGzYhbbv7StO_" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="588" data-original-width="540" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEhRpa47xlPxWsNqUte_9M-2bIkYY4mfBsRN-GY4jm3fsfL5v9Ev8mjfFQ5kTXXD0Ji54srkZzhtCyMWi680U8bd5-6pg4KMuOUw8AJYCkBdDdmzrkCov9if74NdqK0pa_fxCxHfmJMqd_S4NeREHfKyxN8SVCZOfKq1R_dqFs3KQyRgXtDPGzYhbbv7StO_=w367-h400" width="367" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhmW3-hG2727tLsIqO3YrdHqI5tlyq-8qfd0SPywPBy1-mhWgFo4et3l7dzX0e8f88wnmVIQI8pxlqW0ZYY-kfOEU3Eoca4CVb_o8T7UQfAaksHzwtuo4X51gtaFc3XBsLRvBh8efLnvGjZxbqLZkhJTeuoh12rSSGHsQfcWQ1mrWjSgUnnqvoDFgSLDK18" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="523" data-original-width="539" height="389" src="https://blogger.googleusercontent.com/img/a/AVvXsEhmW3-hG2727tLsIqO3YrdHqI5tlyq-8qfd0SPywPBy1-mhWgFo4et3l7dzX0e8f88wnmVIQI8pxlqW0ZYY-kfOEU3Eoca4CVb_o8T7UQfAaksHzwtuo4X51gtaFc3XBsLRvBh8efLnvGjZxbqLZkhJTeuoh12rSSGHsQfcWQ1mrWjSgUnnqvoDFgSLDK18=w400-h389" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">I can see now that it's segmenting too finely. The solution to that (something I know) is to blur the image a little. So I tell it to apply a median filter, and poof again it does it!</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEguztX31UpTlOwI7xWmuEohcLz6rG1Qgu81j9RK4kkcrOMebpyIDBP4EZXJGkP7Bs2xgrfnTr2pklhQuHcDlHSmC6a40henIHhKp8UNCqpOXuZ02iBoyDg0UKJFu7zZSgyRyTxGCV2DIAXd8F9MmZTEIfMHKZSCRURxK-aKJlYgR6Jayi2-5qmrzMM4SEU1" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="616" data-original-width="609" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEguztX31UpTlOwI7xWmuEohcLz6rG1Qgu81j9RK4kkcrOMebpyIDBP4EZXJGkP7Bs2xgrfnTr2pklhQuHcDlHSmC6a40henIHhKp8UNCqpOXuZ02iBoyDg0UKJFu7zZSgyRyTxGCV2DIAXd8F9MmZTEIfMHKZSCRURxK-aKJlYgR6Jayi2-5qmrzMM4SEU1=w395-h400" width="395" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgDyU5Pn8g5mZ8Bo-NySocvXVPUK6fdeyJ8b0-Sg5mFdEW8scpJIKYyrDligZPEx9pHBj_j4uU5UcC-CFWTdB8vMB4Bfw-8B2QaIYWoEWqdKCMyTfjYg44ShvfA-3tR6aIfK5AyQimrmYWRGC2BxovGCmWPM51h957UGs1GmKxaIR1CH9QJ_4Kz4ocugKop" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="587" data-original-width="550" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEgDyU5Pn8g5mZ8Bo-NySocvXVPUK6fdeyJ8b0-Sg5mFdEW8scpJIKYyrDligZPEx9pHBj_j4uU5UcC-CFWTdB8vMB4Bfw-8B2QaIYWoEWqdKCMyTfjYg44ShvfA-3tR6aIfK5AyQimrmYWRGC2BxovGCmWPM51h957UGs1GmKxaIR1CH9QJ_4Kz4ocugKop=w375-h400" width="375" /></a></div><br /></div><div class="separator" style="clear: both; text-align: justify;">Now it's just a matter of dialing in the thresholding correctly...</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhB9_b1rTOUL4y-wkl3FB98OzZW_pwodY6JUZT9obkeJhOSKdP5-IrYdnt9pNOz9a8lpFsV6ESyEgYWW0Led8m9MD44GyeDWRn05GPaFZJaBvN52e9hDaB568Bo1XxzDbkQubY6Q5suoXfP8mX3XJwvFz98FFD1nwS-Y6rfjfpxMpvcYZNWBGV6_cXMgSlY" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="594" data-original-width="595" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEhB9_b1rTOUL4y-wkl3FB98OzZW_pwodY6JUZT9obkeJhOSKdP5-IrYdnt9pNOz9a8lpFsV6ESyEgYWW0Led8m9MD44GyeDWRn05GPaFZJaBvN52e9hDaB568Bo1XxzDbkQubY6Q5suoXfP8mX3XJwvFz98FFD1nwS-Y6rfjfpxMpvcYZNWBGV6_cXMgSlY=w400-h400" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg6VMhIHrq3ZDMHKYN5UwyeWBg1GOrQ47CIcN71vx7zh9DDNMdcwwaEldbbAOQtbCJvN8AS4GJ2xJEYXQYC_vX3KIu1PAwA_mSV1IHC26pRfpbNVaXVWtIreck-SzQvipKa9VxOHu-ha28k3XSAy9LlGxYJqgBHNoz43cPo1i069KI9heiyLHWKaE9pbRAJ" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="504" data-original-width="578" height="349" src="https://blogger.googleusercontent.com/img/a/AVvXsEg6VMhIHrq3ZDMHKYN5UwyeWBg1GOrQ47CIcN71vx7zh9DDNMdcwwaEldbbAOQtbCJvN8AS4GJ2xJEYXQYC_vX3KIu1PAwA_mSV1IHC26pRfpbNVaXVWtIreck-SzQvipKa9VxOHu-ha28k3XSAy9LlGxYJqgBHNoz43cPo1i069KI9heiyLHWKaE9pbRAJ=w400-h349" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhsG3HB_McqgxLz7ScbdVO4_fG-qb9fhBM3s9ttRJb_nsq2TEyFnr8ProqxNlDhBWyc5_nbFmU2Hojm3lSf5W4mZxOs_bcj6I1i-Pox5QKHu_YnsQig1h_dPJ3gONCFZK7umuUqSDCrRmvoIFxPE9Q-d7_LRzFpioau4zpyPDmqBjz5zZfNokCM7uy3zFTO" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="551" data-original-width="602" height="366" src="https://blogger.googleusercontent.com/img/a/AVvXsEhsG3HB_McqgxLz7ScbdVO4_fG-qb9fhBM3s9ttRJb_nsq2TEyFnr8ProqxNlDhBWyc5_nbFmU2Hojm3lSf5W4mZxOs_bcj6I1i-Pox5QKHu_YnsQig1h_dPJ3gONCFZK7umuUqSDCrRmvoIFxPE9Q-d7_LRzFpioau4zpyPDmqBjz5zZfNokCM7uy3zFTO=w400-h366" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjZIUZeEmhe8HZeu_N8Ztj8hn5jzF-6Q1Tm-QEJ5ibfRVNY5IDak5khMYmKNXMI65ZbI-pm8UYUGbHv3SrMgHKUgk_W-FB6QgmADyxlt94LDAuPw7T5VdZD8dxSaDW41DKrtJcQiYolFuj15_ZuSioyFZrHtEuCPTezlAEyKEQjgFJb7DANPF6tj6lQZVgd" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="500" data-original-width="584" height="343" src="https://blogger.googleusercontent.com/img/a/AVvXsEjZIUZeEmhe8HZeu_N8Ztj8hn5jzF-6Q1Tm-QEJ5ibfRVNY5IDak5khMYmKNXMI65ZbI-pm8UYUGbHv3SrMgHKUgk_W-FB6QgmADyxlt94LDAuPw7T5VdZD8dxSaDW41DKrtJcQiYolFuj15_ZuSioyFZrHtEuCPTezlAEyKEQjgFJb7DANPF6tj6lQZVgd=w400-h343" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgniRQhBBp8RpvlGPv59Eeoy7yCgv4EnTocKgK3cNjwbO3PnyhONm0W9KHreBr1jP24ZLDnzlbOSPmHX-wfKlhpAV3_TS8katr8pD8ltopblNBZGm9jlZCNqt0Ol_40jJki3aaixoMwylfntvib1EmN9USzkJ7o9RG4KwciKu9WrkIMjjszrjmaNyJ5bWG-" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="556" data-original-width="608" height="366" src="https://blogger.googleusercontent.com/img/a/AVvXsEgniRQhBBp8RpvlGPv59Eeoy7yCgv4EnTocKgK3cNjwbO3PnyhONm0W9KHreBr1jP24ZLDnzlbOSPmHX-wfKlhpAV3_TS8katr8pD8ltopblNBZGm9jlZCNqt0Ol_40jJki3aaixoMwylfntvib1EmN9USzkJ7o9RG4KwciKu9WrkIMjjszrjmaNyJ5bWG-=w400-h366" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">There we go...</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEibIkK5qXUBbeYRxZbWiLeaXsosG3tCQmw52POzXosXnh0aPRWisRjVheQ7Y_fqMO96PFUAUCG8d6VVEf75obtHe9mDwrrZCvpHWSnHg4hzvN33MgkZqXXOheBFXuRRzgeY1zTvJQhwAsGlhVPNceZeV7qBEeMkaQsSWafZjTJco85CScg9rBvvGTuyHtSh" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="561" data-original-width="583" height="386" src="https://blogger.googleusercontent.com/img/a/AVvXsEibIkK5qXUBbeYRxZbWiLeaXsosG3tCQmw52POzXosXnh0aPRWisRjVheQ7Y_fqMO96PFUAUCG8d6VVEf75obtHe9mDwrrZCvpHWSnHg4hzvN33MgkZqXXOheBFXuRRzgeY1zTvJQhwAsGlhVPNceZeV7qBEeMkaQsSWafZjTJco85CScg9rBvvGTuyHtSh=w400-h386" width="400" /></a></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">It provides some funky labels, but it got 'em!</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgbFE70oi8mfr_0vIF37ReEOFOx6QW9dMl5kKJKf-Syulqc_X_81X9Du8iB3v_6kpSr177nuAz-fQezIMLgNyXGNGC7sPyTP515JnYtlFn80k7P4cgSyavseGLGdNw2MKBT3Q6H7IAlKMa6B9FyE1enkhZlOp0kWAKkl0W1vWNiYQQif8yDSA3e8v8i2Cwn" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="588" data-original-width="534" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEgbFE70oi8mfr_0vIF37ReEOFOx6QW9dMl5kKJKf-Syulqc_X_81X9Du8iB3v_6kpSr177nuAz-fQezIMLgNyXGNGC7sPyTP515JnYtlFn80k7P4cgSyavseGLGdNw2MKBT3Q6H7IAlKMa6B9FyE1enkhZlOp0kWAKkl0W1vWNiYQQif8yDSA3e8v8i2Cwn=w363-h400" width="363" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Please note that it's counting 14 objects. That one penny on the far right is pretty tough to threshold properly so it sees that as two objects. But 1/13 is a score of 92.3%, which is ok and probably at least as good as I am.</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">At the end, I asked it why it saw 910 objects at first instead of 14?</div><div class="separator" style="clear: both; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkp8-0iF_iL8wiIkWJEXqZlwlJTc8wmh3KAfj8xdGF_r4cGw67x1ze9tOD3dGFkVxtfzMWoZP_ZdGVCgXNw-5ZQ6hQYuJnVuRGrlbwp0vQrtMOsVB04EJcLO0KDcYzd2B6jawDpWwpKwLm0tcES1LplS8h3q5A1eudX_MYOw9PH23VyYQDDudZudDJuu_Q/s611/Screenshot%20from%202023-12-11%2016-05-29.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="611" data-original-width="539" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkp8-0iF_iL8wiIkWJEXqZlwlJTc8wmh3KAfj8xdGF_r4cGw67x1ze9tOD3dGFkVxtfzMWoZP_ZdGVCgXNw-5ZQ6hQYuJnVuRGrlbwp0vQrtMOsVB04EJcLO0KDcYzd2B6jawDpWwpKwLm0tcES1LplS8h3q5A1eudX_MYOw9PH23VyYQDDudZudDJuu_Q/w353-h400/Screenshot%20from%202023-12-11%2016-05-29.png" width="353" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div></div></div><div class="separator" style="clear: both; text-align: justify;">Overall it's an impressive show as far as an assistant goes.</div></div></div></div></div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-91509724171580315472023-12-11T14:21:00.000-08:002023-12-11T14:21:07.744-08:00Random DALL-E3 Art<div class="separator" style="clear: both; text-align: justify;">Pile of chocolate mint cookies and raspberry sparkling water:</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhqiGeL1TZJaBbEROZd2Lph3n2kSvvGUHJzCNAoK8C3Aie-pM0j3BLKOCSH4sbmEeKzBfj4krR1dd-ISCKTnwMY2ewOm3EAGUmo65vdYRK1FyOk6IW6uZ9hKPiXIxk-aS-QaaukWffenPkNdYFGtpvURFAe7FnAOeQqLQUE5JGi43S-Yr1GAtT6NBiNQzvx" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1024" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEhqiGeL1TZJaBbEROZd2Lph3n2kSvvGUHJzCNAoK8C3Aie-pM0j3BLKOCSH4sbmEeKzBfj4krR1dd-ISCKTnwMY2ewOm3EAGUmo65vdYRK1FyOk6IW6uZ9hKPiXIxk-aS-QaaukWffenPkNdYFGtpvURFAe7FnAOeQqLQUE5JGi43S-Yr1GAtT6NBiNQzvx=w400-h400" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhiAY6MTk0YK0X8wgFkjLVkV3O05Ib_LpXXt18gE2uBNSQSMvPocHM3qwVTMMiNFINjjfZWKRgehVwBbPYX55oB3WLkymZBhIFxl5Cvee82uzGrlZigKEW3iDLebj6uMyu2jepS6bQuABLNvS8ImnYZ1pKWsqxfuWpkhSFWFccbFnOqC-uBtcDHZbgkMBTE" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1024" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEhiAY6MTk0YK0X8wgFkjLVkV3O05Ib_LpXXt18gE2uBNSQSMvPocHM3qwVTMMiNFINjjfZWKRgehVwBbPYX55oB3WLkymZBhIFxl5Cvee82uzGrlZigKEW3iDLebj6uMyu2jepS6bQuABLNvS8ImnYZ1pKWsqxfuWpkhSFWFccbFnOqC-uBtcDHZbgkMBTE=w400-h400" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjPwZyc94s90fEjdzkD91kbxGO5CUpK9sXhUWRB7sKzrr4gHdDbBOj0Lt2_SXkEAo_GWaLZdNjSafEbjq2PPblpvPfvK8ns8sKbzA4OxmbYX43s0VlngSOpWemhGaY1Rv6Y5KI9fH9hgBeK5K490O2E6c_6BKoE4jzfMHbDHm8svT_J1dEUATjBku52QqhE" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1024" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEjPwZyc94s90fEjdzkD91kbxGO5CUpK9sXhUWRB7sKzrr4gHdDbBOj0Lt2_SXkEAo_GWaLZdNjSafEbjq2PPblpvPfvK8ns8sKbzA4OxmbYX43s0VlngSOpWemhGaY1Rv6Y5KI9fH9hgBeK5K490O2E6c_6BKoE4jzfMHbDHm8svT_J1dEUATjBku52QqhE=w400-h400" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjepdB2CVdZSun5J7TXxMX_cpr5ZjtcNrPtSWu72R5O6uYFR2fbWgEDBib_y0cDOg5GUr6Kkiy8NSdR-oLPlp3NDHbvb2_G1KjN0X6jrB8oW3RIXDG92PdLZM2XNTH0SK0n3hHRh7bPBK3OYO0juyF2mh-4a_cjRJ_Cf6ji2fOk7Y35RnbIHbb1ZDd2JfCc" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1024" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEjepdB2CVdZSun5J7TXxMX_cpr5ZjtcNrPtSWu72R5O6uYFR2fbWgEDBib_y0cDOg5GUr6Kkiy8NSdR-oLPlp3NDHbvb2_G1KjN0X6jrB8oW3RIXDG92PdLZM2XNTH0SK0n3hHRh7bPBK3OYO0juyF2mh-4a_cjRJ_Cf6ji2fOk7Y35RnbIHbb1ZDd2JfCc=w400-h400" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhJM1BRAwkJWWBinWAZ0_VCg0_6_Mz6bbt-N8Ke6-XxW7ZddZO7G22IHNGqudQLHHz1BT8jNkrp_m0WtMLl4o1wj7VsC170UlMD5aZjLxnF4WHoSJmFGT1KPq70bZBiA5q9IFGdT_afSKy-R_v-XEfgwEqHxIJTcCTyNkZ9ukhQfCEdhcg2pPd6NOBtFSZ2" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEhJM1BRAwkJWWBinWAZ0_VCg0_6_Mz6bbt-N8Ke6-XxW7ZddZO7G22IHNGqudQLHHz1BT8jNkrp_m0WtMLl4o1wj7VsC170UlMD5aZjLxnF4WHoSJmFGT1KPq70bZBiA5q9IFGdT_afSKy-R_v-XEfgwEqHxIJTcCTyNkZ9ukhQfCEdhcg2pPd6NOBtFSZ2=w400-h229" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhUjHjr0QzNGRfEq33hWih6iAVsERoMHhXeDX7Vo2wjK3JUhUMvrlQB4mdR-q-9e5-dwG_wifyNV0_SO0Lxdq7sQoh2NIZGr-ZmU8KV-xk26-jVpdtSYDRz9TdZv1cPMHDkQh0JEsg3FhwA7uAXGa6VCPYbnmm4BVS95UnohUh_oQfZb18WmRvmDN6SPOrj" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" src="https://blogger.googleusercontent.com/img/a/AVvXsEhUjHjr0QzNGRfEq33hWih6iAVsERoMHhXeDX7Vo2wjK3JUhUMvrlQB4mdR-q-9e5-dwG_wifyNV0_SO0Lxdq7sQoh2NIZGr-ZmU8KV-xk26-jVpdtSYDRz9TdZv1cPMHDkQh0JEsg3FhwA7uAXGa6VCPYbnmm4BVS95UnohUh_oQfZb18WmRvmDN6SPOrj=w400-h229" width="400" /></a></div><br /><br /></div></div></div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-72041144567259083012023-12-05T20:00:00.000-08:002023-12-05T20:00:07.697-08:00Some additional Murmurations<div style="text-align: left;"> <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgAZxNBaRnAmQpyi7pdvZHgatyZPjvGhtf8syORcxPnNl_UfKV7zUTFTt3RfEMfA9SHPckhoaEU9KBdXDHI5gu9xkUvTF_MmME6_dm1VCXQR6eJVsVsxLy2_gholjlNGzx3fwDBD2nJszTFfpA9nBgJxsxua0OD5uwuTdbOUuSURUT3dB2afVL24-o6sfaD" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEgAZxNBaRnAmQpyi7pdvZHgatyZPjvGhtf8syORcxPnNl_UfKV7zUTFTt3RfEMfA9SHPckhoaEU9KBdXDHI5gu9xkUvTF_MmME6_dm1VCXQR6eJVsVsxLy2_gholjlNGzx3fwDBD2nJszTFfpA9nBgJxsxua0OD5uwuTdbOUuSURUT3dB2afVL24-o6sfaD=w400-h225" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiR56yyHNDarMuwNKul-d_LqTxcEqAl_2Zyn_XxbmJQCEIieLIZQkqZOvyMr6tfaelzQeRAnu9VsZYc7En-rZ4JjAsy2vjHWEqMUgmibVuOjFRo7CIAok0laYviKp-ahEdwVWZ-ktOZqad2h7pnPqjaH9iQkwrg08KpJXcNgCssoO_YHNXv1xtyblbgDOTY" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEiR56yyHNDarMuwNKul-d_LqTxcEqAl_2Zyn_XxbmJQCEIieLIZQkqZOvyMr6tfaelzQeRAnu9VsZYc7En-rZ4JjAsy2vjHWEqMUgmibVuOjFRo7CIAok0laYviKp-ahEdwVWZ-ktOZqad2h7pnPqjaH9iQkwrg08KpJXcNgCssoO_YHNXv1xtyblbgDOTY=w400-h225" width="400" /></a></div><br /></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiYj9c6G7OsvjSmItRbOGG2UHE0eCKV72zfWNw68UE4qWX4F5vi73mvGC7a9XLr-zePM4FjzdnLZ2FpxUA2vDPn5SY_lOHRC3iKFLlA4IN5IUWoNHF7IZbb1mXS4WWVBaWCwWbzPOHMd_ga54x2TZlkPh6XOq1yI-VIS-l-PxauYW1uq72ft0RZkitE_Uhw" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEiYj9c6G7OsvjSmItRbOGG2UHE0eCKV72zfWNw68UE4qWX4F5vi73mvGC7a9XLr-zePM4FjzdnLZ2FpxUA2vDPn5SY_lOHRC3iKFLlA4IN5IUWoNHF7IZbb1mXS4WWVBaWCwWbzPOHMd_ga54x2TZlkPh6XOq1yI-VIS-l-PxauYW1uq72ft0RZkitE_Uhw=w400-h225" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjLSh4TLdTI8hxsHWbnpJvZgtId1KzqVizF67ccaNmya8Pxu3g8ssg0QsLJqTP6P2k79j5bDD_YvXbMNemWC9EFPgr0Sis6Vco-0ywuk-5ZSPq3KFcvNjQ_fpONoAibWF9MU7UzryqYeyIxYC0FUlfcTSpS2rmucyXIUPz0_3pZtBY8Imy4-5ombMADJgXl" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEjLSh4TLdTI8hxsHWbnpJvZgtId1KzqVizF67ccaNmya8Pxu3g8ssg0QsLJqTP6P2k79j5bDD_YvXbMNemWC9EFPgr0Sis6Vco-0ywuk-5ZSPq3KFcvNjQ_fpONoAibWF9MU7UzryqYeyIxYC0FUlfcTSpS2rmucyXIUPz0_3pZtBY8Imy4-5ombMADJgXl=w400-h225" width="400" /></a></div><br /><br /></div><br /><br /></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-72673023028226595132023-12-04T18:30:00.000-08:002023-12-04T19:31:07.894-08:00Murmurations<div style="text-align: justify;">Apologies for not including my murmuration project on this blog. Every year from October through April, the Sand Hill cranes call my location in southeastern arizona usa, home. They fly in huge 'V' shaped murmurations, and for years I've been wanting to 'do' something with this. It's a beautiful sight, and I'd like to understand better how they fly so well together and how they communicate between one another.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Anyhow I've been stuck for several years, but a couple months ago I stumbled across work being done by Doris Mitsch. Her work showed me a clear path onto 'something' but I'm yet to be exactly sure what. At the moment, I'm just collecting data and playing with my image processing code that generates long exposure images of the videos that I'm taking of these birds in flight.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">There seem to be two primary modes of flight: straight ahead in their typical 'V' shapes, and a rotating action they do mid-day that I'm calling 'meetings' or conferences. During these meetings they fly in tight circles and talk very loudly. After a few turns a group breaks off and everyone else follows them back into their 'V' formations. Fun to watch, and currently my software is generating these long exposure images.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Here are some examples, and there are some interesting videos on my youtube channel.</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj_PDkUOoeCJWA6y_GTvdIT7nk9KALerVoz28Pk-UaFQ9SLvSlsOFYKy3VWh8qqyuRyBRErGiyfAW81u9i-LPfkfzKijxoetwZW6x_ds9c2j6n_cbV47ijq38fe7RqfH9YCbs8kXaqkVYj04azCIoD-i99S6HJJWKYs2gZTh35S3MJ9t_XrLjPiGu_n3nKy" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEj_PDkUOoeCJWA6y_GTvdIT7nk9KALerVoz28Pk-UaFQ9SLvSlsOFYKy3VWh8qqyuRyBRErGiyfAW81u9i-LPfkfzKijxoetwZW6x_ds9c2j6n_cbV47ijq38fe7RqfH9YCbs8kXaqkVYj04azCIoD-i99S6HJJWKYs2gZTh35S3MJ9t_XrLjPiGu_n3nKy=w400-h225" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">there were a couple of crows or ravens flying around too.</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiATWfKExAB_ozKE3UL1Dudw3l-YCoQTHXEYJc1TonyOc24X41vFrIgtvqtbVZ6xAQjqXygjFcRfGACvQnyuzK8i1WCzK_hqPtn1sTRIAM92U6UOocRo1hiJPhMBRgusKWwHWQnUXIGzfY5jMy6JDQFzGS7F3DWH5YsuPySM1ZJ5yCWO1N9NxI0H1C01AlW" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEiATWfKExAB_ozKE3UL1Dudw3l-YCoQTHXEYJc1TonyOc24X41vFrIgtvqtbVZ6xAQjqXygjFcRfGACvQnyuzK8i1WCzK_hqPtn1sTRIAM92U6UOocRo1hiJPhMBRgusKWwHWQnUXIGzfY5jMy6JDQFzGS7F3DWH5YsuPySM1ZJ5yCWO1N9NxI0H1C01AlW=w400-h225" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEisYvdWlqVrDLYaCvjWinTK1jJxNT7Gq1CYUCvdHkajlLkPqas1Sa0jdLI_dhtMjZZorP4qzc7oPB1A8Y_PYA__KezYspiEAg9MmjabApl06G7v5Iy_KJctwB_p5D4V55ORR4vnU2PSX8ohBuJU_4_sosq5BXg6qDFuH2yCsiJ-7bWnND3zmzI7qMDkD226" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1080" data-original-width="1920" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEisYvdWlqVrDLYaCvjWinTK1jJxNT7Gq1CYUCvdHkajlLkPqas1Sa0jdLI_dhtMjZZorP4qzc7oPB1A8Y_PYA__KezYspiEAg9MmjabApl06G7v5Iy_KJctwB_p5D4V55ORR4vnU2PSX8ohBuJU_4_sosq5BXg6qDFuH2yCsiJ-7bWnND3zmzI7qMDkD226=w400-h225" width="400" /></a></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhtxOG_QQsVVdDSO3CMtsXYzff7ii61YOKBs9Iln_6oRyA-Y7V4gtaejpZIT7Or4j4Dfno1flbu3ylNz02oVQZKQIZX3Z00oH7lZEIizwJUrhBH9W36gS0DIYrbjRxaadnu2_vGK_nJEXa8TPkucZgVLqzVD9JhkH8MLR6g44ZVzvFUO-PowNISLmPDi1Jf" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEhtxOG_QQsVVdDSO3CMtsXYzff7ii61YOKBs9Iln_6oRyA-Y7V4gtaejpZIT7Or4j4Dfno1flbu3ylNz02oVQZKQIZX3Z00oH7lZEIizwJUrhBH9W36gS0DIYrbjRxaadnu2_vGK_nJEXa8TPkucZgVLqzVD9JhkH8MLR6g44ZVzvFUO-PowNISLmPDi1Jf=w400-h225" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiqPu1YqgSj4Z1lGW0VYGYdJ0-al-_i-q1CNXOJ0cA3YspMpVFbDvkCsKBdMh6CDt5muFeJMkN6TiOCjlE5qF5eLJ24W3b808XLNj76tLZPiAL6rkC1MZ2EVCN6d_KaPfok8sosnyIFv5RSxxoE6ctugI1_TKqiIXpRR26mP5LHDgXQyomh1tesKyleJNb3" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEiqPu1YqgSj4Z1lGW0VYGYdJ0-al-_i-q1CNXOJ0cA3YspMpVFbDvkCsKBdMh6CDt5muFeJMkN6TiOCjlE5qF5eLJ24W3b808XLNj76tLZPiAL6rkC1MZ2EVCN6d_KaPfok8sosnyIFv5RSxxoE6ctugI1_TKqiIXpRR26mP5LHDgXQyomh1tesKyleJNb3=w400-h225" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjwZHXxaag_MRbZ7XXlaxDNC0zQtKlS-XN-RCeM2flM_TzyPuey_dBVeHVO8CCiyhmrzbWYucm0pty0rhSkJzIWjp7e94s26jJpEpRBqK2XDVPw4ns770dH2Xt5QHY19Wqdtr9j8Vzgl2-IkEbP9dn_NUdvF7vOhBY3Ln928xOp6hecwYuj8qx-uEsGtagJ" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEjwZHXxaag_MRbZ7XXlaxDNC0zQtKlS-XN-RCeM2flM_TzyPuey_dBVeHVO8CCiyhmrzbWYucm0pty0rhSkJzIWjp7e94s26jJpEpRBqK2XDVPw4ns770dH2Xt5QHY19Wqdtr9j8Vzgl2-IkEbP9dn_NUdvF7vOhBY3Ln928xOp6hecwYuj8qx-uEsGtagJ=w400-h225" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgw4SKFrcSpTz7rPAlc1LQEQr6JJugonsqs08WOnR-ueGjtx8y6aBCWjSejz_KQtKn8wWcIwCyvBUuO2o0n10ANGxQcrmdPhgx14z4MLAVbj-8h9EbCs4mz_UjTJ8LMPeWIz8JhH6a0-EKFtHFd9S7apfCoXFG9Un33pl9OPaWqDUE3KthzD0fYbo51w-mq/s1300/Screenshot%20from%202023-12-04%2020-20-24.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="731" data-original-width="1300" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgw4SKFrcSpTz7rPAlc1LQEQr6JJugonsqs08WOnR-ueGjtx8y6aBCWjSejz_KQtKn8wWcIwCyvBUuO2o0n10ANGxQcrmdPhgx14z4MLAVbj-8h9EbCs4mz_UjTJ8LMPeWIz8JhH6a0-EKFtHFd9S7apfCoXFG9Un33pl9OPaWqDUE3KthzD0fYbo51w-mq/w400-h225/Screenshot%20from%202023-12-04%2020-20-24.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">wiggly paths:</div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhEkXA2RLRI49XeHw-D0Zy3fb4AlLAVE38UO_px0U7UghEGy71NtupoOomrs-muq2HJJWRqEmTM8XQVh253TcI5HWcur-QflILdP-eBNnbz3oQmWDv1gJhuFQ-9jzPmANGJ5IBPwNpGhaKCfXF3k6s02dab3dZTBjKDPeTAlymhHsk4MS38q1ADzBGc9nYn" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="691" data-original-width="850" height="325" src="https://blogger.googleusercontent.com/img/a/AVvXsEhEkXA2RLRI49XeHw-D0Zy3fb4AlLAVE38UO_px0U7UghEGy71NtupoOomrs-muq2HJJWRqEmTM8XQVh253TcI5HWcur-QflILdP-eBNnbz3oQmWDv1gJhuFQ-9jzPmANGJ5IBPwNpGhaKCfXF3k6s02dab3dZTBjKDPeTAlymhHsk4MS38q1ADzBGc9nYn=w400-h325" width="400" /></a></div><br /></div></div><div class="separator" style="clear: both; text-align: center;"><br /></div></div>I've used chatGPT to give me some suggestions for research topics, and this is what it came up with (I asked for 20 topics):</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">--------------------------------------</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><div class="separator" style="clear: both;">User</div><div class="separator" style="clear: both;">you are an expert in the murmurations of sand hill cranes. Please recommend twenty lines of scientific investigation about the murmurations of sand hill cranes</div><div class="separator" style="clear: both;">ChatGPT</div><div class="separator" style="clear: both;">Studying the murmurations of Sandhill Cranes is a fascinating and complex field. Here are twenty lines of scientific investigation that can be explored:</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Behavioral Patterns: Analyzing the specific behaviors exhibited during murmurations and understanding their social or survival functions.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Communication Mechanisms: Investigating how cranes communicate during these gatherings, focusing on both auditory and visual signals.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Environmental Influences: Studying how different environmental conditions, such as weather or habitat changes, affect murmuration patterns.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Migration and Murmurations: Understanding how migratory patterns influence murmuration behaviors, timings, and locations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Population Dynamics: Examining the relationship between crane population sizes and the scale or frequency of murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Predator-Prey Interactions: Investigating if and how murmurations function as a defensive mechanism against predators.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Nutritional Aspects: Understanding the feeding behaviors and nutritional intake during murmuration periods.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Breeding and Murmurations: Studying the role of murmurations in mating and breeding behaviors of Sandhill Cranes.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Social Hierarchies: Analyzing if there are any observable social structures or hierarchies within murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Age-Related Behaviors: Examining differences in murmuration participation or behavior based on the age of the cranes.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Genetic Factors: Investigating whether genetic factors influence participation in, or the behavior during, murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Physiological Stress Responses: Studying the physiological impacts, such as stress responses, of cranes during large gatherings.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Aerodynamic Aspects: Analyzing the flight patterns and aerodynamics involved in murmuration formations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Impact of Human Activities: Assessing how human activities, like agriculture or urbanization, impact crane murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Conservation Implications: Understanding how knowledge of murmurations can contribute to effective conservation strategies for Sandhill Cranes.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Disease Transmission: Investigating the role of murmurations in the spread or control of diseases among crane populations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Acoustic Analysis: Studying the sounds made during murmurations and their potential roles or meanings.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Technological Monitoring: Utilizing modern technologies like drones or satellite tracking to study murmurations in greater detail.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Comparative Studies: Comparing murmuration behaviors of Sandhill Cranes with other species to identify unique or common patterns.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Climate Change Impact: Researching how changing climate conditions are affecting the murmuration behaviors of Sandhill Cranes.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Each of these lines of inquiry could contribute significantly to our understanding of the complex social and ecological aspects of Sandhill Cranes' murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">-----------------</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">I then went on:</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">-----------------------</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;"><div class="separator" style="clear: both;">User</div><div class="separator" style="clear: both;">I'm particularly interested in the aerodynamic aspects -- will you please expand upon this and present me with twenty suggestions for research projects</div><div class="separator" style="clear: both;">ChatGPT</div><div class="separator" style="clear: both;">Focusing on the aerodynamic aspects of Sandhill Crane murmurations, here are twenty research project suggestions that delve into various facets of this intriguing phenomenon:</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Formation Flight Analysis: Study the aerodynamic benefits of different flight formations during murmurations, such as V-formations or echelons.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Energy Efficiency: Investigate how different murmuration patterns affect the energy expenditure of individual cranes during flight.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Wingtip Vortex Interactions: Analyze the interactions of wingtip vortices between individual cranes in a murmuration and how this influences flight efficiency.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Maneuverability in Murmurations: Study how cranes maneuver within a murmuration, focusing on the aerodynamic mechanisms that allow for quick and coordinated changes in direction.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Wind Tunnel Experiments: Conduct wind tunnel experiments with crane models to study the aerodynamic forces at play during murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Computational Fluid Dynamics (CFD) Models: Develop CFD models to simulate the airflow around and within crane murmurations, providing insights into the aerodynamic interactions.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Flight Speed and Aerodynamics: Examine how flight speed affects the aerodynamic efficiency of cranes during murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Altitude Effects on Flight Dynamics: Research how different altitudes, with varying air densities, affect the aerodynamics of crane murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Thermal Soaring and Murmurations: Investigate the role of thermal currents in the aerodynamics of murmurations, especially in relation to soaring behavior.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Feather Morphology and Aerodynamics: Study the relationship between feather morphology of Sandhill Cranes and their aerodynamic performance during murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Aerodynamic Noise Analysis: Analyze the aerodynamic noise generated during murmurations and its potential communication roles.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Impact of Weather Conditions: Investigate how varying weather conditions, like wind speed and direction, affect the aerodynamics of murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Energy Transfer Within Flocks: Examine how energy is transferred between individuals in a murmuration, particularly through aerodynamic interactions.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Landing and Takeoff Dynamics: Study the aerodynamics involved in the coordinated takeoff and landing behaviors during murmurations.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Aerodynamic Benefits of Synchronization: Research how synchronized movements in murmurations contribute to aerodynamic efficiency.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Individual vs. Group Aerodynamics: Compare the aerodynamics of solitary crane flight to that within a murmuration to understand the benefits of group flight.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Stress Analysis on Wings during Murmurations: Conduct biomechanical studies to understand the stress and strain on crane wings during different murmuration maneuvers.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Use of Drone Technology: Employ drones to mimic crane flight patterns in murmurations and study the resulting aerodynamic data.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Evolutionary Aspects of Aerodynamic Efficiency: Explore the evolutionary development of aerodynamic efficiency in murmurations as a survival trait.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Impact of Age and Physical Condition: Research how the age and physical condition of individual cranes affect their aerodynamic roles and capabilities within a murmuration.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Each of these projects would contribute to a deeper understanding of the complex aerodynamics involved in Sandhill Crane murmurations, providing insights into both the behavior of these birds and the broader principles of flight in flocking birds.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">------------------------------</div></div></div></div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-57728377084117008842023-12-04T16:04:00.000-08:002023-12-04T16:04:35.862-08:00Images Generated by DALL-E3 using a quote from Alan Watts"... your skin doesn't separate you from the world it's a bridge through which the external world flows into you and you flow into it just for example as a whirlpool in water you could say because you have a skin you have a definite shape you have a definite form the whirlpool is a definite form but no water stays put in it the whirlpool is something the stream is doing and exactly the same way the whole universe is doing each one of us" - Alan Watts<div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjzpwVpY-99FAAcHcAk4gOstPc4ylf-sfH-ggtYALoEjj5FME5x7OtEv5BRmhZlZkJaukhM8lADuEsysEJ7vmu4BSAF9lopTky3EtbNB2OjtIGWdDZfjoQs2CZhS-RH2iKpdduhAO2fD12Y4WBYNJ4xGfs3nc-j5byhiNAjTv-2zdrZeylMqTELl6qoV3Lb" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1024" data-original-width="1792" height="229" 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href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3XYTAmXgSBvV0MWUVlOE8QS16Fj-HK7OOAfEqi86BLy8H5uVb1Mg8hlJ3xKG3Gs8Og93CDawODd57JWYeGhyMmqrXiaq2dADnX4JiAhl3URCjibG7llhfGhI-qF9A_IjpAndwzj_ZpPF91ArMZVry53NXW1bQ8Si8CoR-VnTC0zD47XLPRxvzMib7gJOw/s1024/DALL%C2%B7E%202023-11-28%2014.58.09%20-%20A%20surreal,%20artistic%20representation%20of%20rock%20balancing%20structures,%20including%20at%20least%20two%20rock%20arches,%20and%20a%20deserted%20desert%20outpost,%20all%20set%20in%20outer%20s.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1024" data-original-width="1024" height="400" 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/></a></div><br /></div><br /></div></div></div></div></div></div></div></div><p></p>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-36022395682574974732023-09-20T18:00:00.003-07:002023-09-20T19:26:37.487-07:00chatGPT's Superpower is *** Self-checking ***<div style="text-align: left;"><div class="group w-full text-token-text-primary border-b border-black/10 gizmo:border-0 dark:border-gray-900/50 gizmo:dark:border-0 gizmo:bg-transparent dark:bg-gray-800 gizmo:dark:bg-transparent" data-testid="conversation-turn-2" style="--avatar-color: #AB68FF; --tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-color: rgba(0, 0, 0, 0.1); border-image: initial; border-style: solid; border-width: 0px 0px 1px; box-sizing: border-box; color: var(--text-primary); font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 14px; width: 1172px;"><div class="p-4 justify-center text-base md:gap-6 md:py-6 m-auto" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-size: 1rem; gap: 1.5rem; justify-content: center; line-height: 1.5rem; margin: auto; padding: 1.5rem 1rem;"><div class="flex flex-1 gap-4 text-base mx-auto md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl }" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex: 1 1 0%; font-size: 1rem; gap: 1.5rem; line-height: 1.5rem; margin-left: auto; margin-right: auto; max-width: 48rem;"><div class="flex-shrink-0 flex flex-col relative items-end" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-end; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-shrink: 0; position: relative;"><div style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><div class="relative flex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; position: relative;"><img alt="User" class="rounded-sm" data-nimg="1" decoding="async" height="36" loading="lazy" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=96&q=75" srcset="/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=48&q=75 1x, /_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=96&q=75 2x" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-radius: 0.125rem; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: transparent; display: block; height: auto; max-width: 100%; vertical-align: middle;" width="36" /></div></div><div class="text-xs flex items-center justify-center gap-1 absolute left-0 top-2 -ml-4 -translate-x-full invisible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: -100%; --tw-translate-y: 0; align-items: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; font-size: 0.75rem; gap: 0.25rem; justify-content: center; left: 0px; line-height: 1rem; margin-left: -1rem; position: absolute; top: 0.5rem; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: hidden;"><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0 tabular-nums" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-numeric-spacing: tabular-nums; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; flex-grow: 1; flex-shrink: 0; font-variant-numeric: var(--tw-ordinal) var(--tw-slashed-zero) var(--tw-numeric-figure) var(--tw-numeric-spacing) var(--tw-numeric-fraction);"></span><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; position: relative; width: calc(100% - 115px);"><div class="flex flex-grow flex-col gap-3 max-w-full" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-grow: 1; gap: 0.75rem; max-width: 100%;"><div class="min-h-[20px] flex flex-col items-start gap-3 overflow-x-auto whitespace-pre-wrap break-words" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-start; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; min-height: 20px; overflow-wrap: break-word; overflow-x: auto; white-space: pre-wrap;"><div class="empty:hidden" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;">you are an expert at mathematics. generate an equation of a circle that goes through the points (0,0), (0.5,0.5), (1,0)</div></div></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 100%; --tw-translate-y: 0; align-self: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(172,172,190,var(--tw-text-opacity)); display: flex; gap: 0.25rem; justify-content: center; margin-top: 0px; padding-left: 0.5rem; position: absolute; right: 0px; top: 0px; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: visible;"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400 md:invisible md:group-hover:visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem; visibility: hidden;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div><div class="flex justify-between lg:block" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; justify-content: space-between;"></div></div></div></div></div><div class="group w-full text-token-text-primary border-b border-black/10 gizmo:border-0 dark:border-gray-900/50 gizmo:dark:border-0 bg-gray-50 gizmo:bg-transparent dark:bg-[#444654] gizmo:dark:bg-transparent" data-testid="conversation-turn-3" style="--avatar-color: #AB68FF; --tw-bg-opacity: 1; --tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-color: rgba(0, 0, 0, 0.1); border-image: initial; border-style: solid; border-width: 0px 0px 1px; box-sizing: border-box; color: var(--text-primary); font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 14px; width: 1172px;"><div class="p-4 justify-center text-base md:gap-6 md:py-6 m-auto" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-size: 1rem; gap: 1.5rem; justify-content: center; line-height: 1.5rem; margin: auto; padding: 1.5rem 1rem;"><div class="flex flex-1 gap-4 text-base mx-auto md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl }" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex: 1 1 0%; font-size: 1rem; gap: 1.5rem; line-height: 1.5rem; margin-left: auto; margin-right: auto; max-width: 48rem;"><div class="flex-shrink-0 flex flex-col relative items-end" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-end; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-shrink: 0; position: relative;"><div style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><div class="relative p-1 rounded-sm h-9 w-9 text-white flex items-center justify-center" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; align-items: center; background-color: #ab68ff; border-radius: 0.125rem; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(255,255,255,var(--tw-text-opacity)); display: flex; height: 36px; justify-content: center; padding: 0.25rem; position: relative; width: 36px;"><svg class="icon-md" fill="none" height="41" role="img" viewbox="0 0 41 41" width="41" xmlns="http://www.w3.org/2000/svg"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 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36.8382 27.0432 35.9743 28.1759C35.1103 29.3086 33.9415 30.1717 32.6047 30.6641C32.6047 30.5947 32.6047 30.4733 32.6047 30.3889V21.188C32.6066 20.9586 32.5474 20.7328 32.4332 20.5338C32.319 20.3348 32.154 20.1698 31.955 20.0556ZM35.3055 15.0128C35.2464 14.9765 35.1431 14.9142 35.069 14.8717L27.1045 10.2712C26.906 10.1554 26.6803 10.0943 26.4504 10.0943C26.2206 10.0943 25.9948 10.1554 25.7963 10.2712L16.0726 15.8858V11.9982C16.0715 11.9783 16.0753 11.9585 16.0837 11.9405C16.0921 11.9225 16.1048 11.9068 16.1207 11.8949L24.1719 7.25025C25.4053 6.53903 26.8158 6.19376 28.2383 6.25482C29.6608 6.31589 31.0364 6.78077 32.2044 7.59508C33.3723 8.40939 34.2842 9.53945 34.8334 10.8531C35.3826 12.1667 35.5464 13.6095 35.3055 15.0128ZM14.2424 21.9419L10.8752 19.9981C10.8576 19.9893 10.8423 19.9763 10.8309 19.9602C10.8195 19.9441 10.8122 19.9254 10.8098 19.9058V10.6071C10.8107 9.18295 11.2173 7.78848 11.9819 6.58696C12.7466 5.38544 13.8377 4.42659 15.1275 3.82264C16.4173 3.21869 17.8524 2.99464 19.2649 3.1767C20.6775 3.35876 22.0089 3.93941 23.1034 4.85067C23.0427 4.88379 22.937 4.94215 22.8668 4.98473L14.9024 9.58517C14.7025 9.69878 14.5366 9.86356 14.4215 10.0626C14.3065 10.2616 14.2466 10.4877 14.2479 10.7175L14.2424 21.9419ZM16.071 17.9991L20.4018 15.4978L24.7325 17.9975V22.9985L20.4018 25.4983L16.071 22.9985V17.9991Z" fill="currentColor"></path></svg></div></div><div class="text-xs flex items-center justify-center gap-1 absolute left-0 top-2 -ml-4 -translate-x-full invisible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: -100%; --tw-translate-y: 0; align-items: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; font-size: 0.75rem; gap: 0.25rem; justify-content: center; left: 0px; line-height: 1rem; margin-left: -1rem; position: absolute; top: 0.5rem; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: hidden;"><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0 tabular-nums" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-numeric-spacing: tabular-nums; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; flex-grow: 1; flex-shrink: 0; font-variant-numeric: var(--tw-ordinal) var(--tw-slashed-zero) var(--tw-numeric-figure) var(--tw-numeric-spacing) var(--tw-numeric-fraction);"></span><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; position: relative; width: calc(100% - 115px);"><div class="flex flex-grow flex-col gap-3 max-w-full" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-grow: 1; gap: 0.75rem; max-width: 100%;"><div class="min-h-[20px] flex flex-col items-start gap-3 overflow-x-auto whitespace-pre-wrap break-words" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-start; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; min-height: 20px; overflow-wrap: break-word; overflow-x: auto; white-space: pre-wrap;"><div class="markdown prose w-full break-words dark:prose-invert light" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-prose-body: #374151; --tw-prose-bold: #111827; --tw-prose-bullets: #d1d5db; --tw-prose-captions: #6b7280; --tw-prose-code: #111827; --tw-prose-counters: #6b7280; --tw-prose-headings: #111827; --tw-prose-hr: #e5e7eb; --tw-prose-invert-body: #d1d5db; --tw-prose-invert-bold: #fff; --tw-prose-invert-bullets: #4b5563; --tw-prose-invert-captions: #9ca3af; --tw-prose-invert-code: #fff; --tw-prose-invert-counters: #9ca3af; --tw-prose-invert-headings: #fff; --tw-prose-invert-hr: #374151; --tw-prose-invert-lead: #9ca3af; --tw-prose-invert-links: #fff; --tw-prose-invert-pre-bg: rgba(0,0,0,0.5); --tw-prose-invert-pre-code: #d1d5db; --tw-prose-invert-quote-borders: #374151; --tw-prose-invert-quotes: #f3f4f6; --tw-prose-invert-td-borders: #374151; --tw-prose-invert-th-borders: #4b5563; --tw-prose-lead: #4b5563; --tw-prose-links: #111827; --tw-prose-pre-bg: #1f2937; --tw-prose-pre-code: #e5e7eb; --tw-prose-quote-borders: #e5e7eb; --tw-prose-quotes: #111827; --tw-prose-td-borders: #e5e7eb; --tw-prose-th-borders: #d1d5db; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: var(--tw-prose-body); font-size: 1rem; line-height: 1.75; max-width: none; overflow-wrap: break-word; width: 653px;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px 0px 1.25em;">To determine the equation of a circle that goes through the given three points, we can use the general equation of a circle:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(x - h)^2 + (y - k)^2 = r^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Where <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(h, k)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mpunct" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">,</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.1667em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span> is the center of the circle and <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.4306em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span></span></span></span></span> is its radius.</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Plugging in the given points:</p><ol style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-reset: list-number 0; display: flex; flex-direction: column; list-style: none; margin: 1.25em 0px; padding: 0px;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span><span class="mpunct" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">,</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.1667em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span> gives:
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">h^2 + k^2 = r^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span>
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">\implies r^2 - h^2 - k^2 = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.549em; vertical-align: -0.024em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">⟹</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span></span></span> ... (i)</p></li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(0.5, 0.5)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mpunct" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">,</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.1667em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span> gives:
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(0.5 - h)^2 + (0.5 - k)^2 = r^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span>
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">\implies r^2 - h^2 + h - k^2 + k = 0.5</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.549em; vertical-align: -0.024em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">⟹</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.7778em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span></span></span></span></span> ... (ii)</p></li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(1,0)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">1</span><span class="mpunct" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">,</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.1667em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span> gives:
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(1 - h)^2 + k^2 = r^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">1</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span>
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">\implies r^2 - h^2 + 2h - k^2 - 1 = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.549em; vertical-align: -0.024em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">⟹</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.7778em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">1</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span></span></span> ... (iii)</p></li></ol><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Now, subtracting (i) from (ii) and (iii) gives:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">h + k = 0.5</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.7778em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span></span></span></span></span> ... (iv)</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">2h - 1 = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.7778em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">1</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span></span></span>
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">\implies h = 0.5</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.549em; vertical-align: -0.024em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">⟹</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span></span></span></span></span> ... (v)</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Plugging (v) into (iv) gives:
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">k = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">The center of the circle is (0.5, 0).</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Plugging <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">h = 0.5</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">h</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span></span></span></span></span> and <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">k = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03148em;">k</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span></span></span> into (i) gives:
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">r^2 = 0.25</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span></span></span></span></span>
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">\implies r = 0.5</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.549em; vertical-align: -0.024em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">⟹</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.4306em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.02778em;">r</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Thus, the equation of the circle is:
<span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(x - 0.5)^2 + y^2 = 0.25</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">or equivalently:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(x - 0.5)^2 + y^2 = (0.5)^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span></p></div></div></div><div class="flex justify-between lg:block" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; justify-content: space-between;"><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 100%; --tw-translate-y: 0; align-self: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(172,172,190,var(--tw-text-opacity)); display: flex; gap: 0.25rem; justify-content: center; margin-top: 0px; padding-left: 0.5rem; position: absolute; right: 0px; top: 0px; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: visible;"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; display: flex; font-family: inherit; font-size: 16px; font-weight: inherit; gap: 0.5rem; line-height: inherit; margin: 0px 0px 0px auto; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect height="4" rx="1" ry="1" width="8" x="8" y="2"></rect></svg></button><div class="flex gap-1" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; gap: 0.25rem;"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div></div><div class="group w-full text-token-text-primary border-b border-black/10 gizmo:border-0 dark:border-gray-900/50 gizmo:dark:border-0 gizmo:bg-transparent dark:bg-gray-800 gizmo:dark:bg-transparent" data-testid="conversation-turn-4" style="--avatar-color: #AB68FF; --tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-color: rgba(0, 0, 0, 0.1); border-image: initial; border-style: solid; border-width: 0px 0px 1px; box-sizing: border-box; color: var(--text-primary); font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 14px; width: 1172px;"><div class="p-4 justify-center text-base md:gap-6 md:py-6 m-auto" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-size: 1rem; gap: 1.5rem; justify-content: center; line-height: 1.5rem; margin: auto; padding: 1.5rem 1rem;"><div class="flex flex-1 gap-4 text-base mx-auto md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl }" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex: 1 1 0%; font-size: 1rem; gap: 1.5rem; line-height: 1.5rem; margin-left: auto; margin-right: auto; max-width: 48rem;"><div class="flex-shrink-0 flex flex-col relative items-end" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-end; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-shrink: 0; position: relative;"><div style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><div class="relative flex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; position: relative;"><img alt="User" class="rounded-sm" data-nimg="1" decoding="async" height="36" loading="lazy" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=96&q=75" srcset="/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=48&q=75 1x, /_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=96&q=75 2x" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-radius: 0.125rem; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: transparent; display: block; height: auto; max-width: 100%; vertical-align: middle;" width="36" /></div></div><div class="text-xs flex items-center justify-center gap-1 absolute left-0 top-2 -ml-4 -translate-x-full invisible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: -100%; --tw-translate-y: 0; align-items: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; font-size: 0.75rem; gap: 0.25rem; justify-content: center; left: 0px; line-height: 1rem; margin-left: -1rem; position: absolute; top: 0.5rem; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: hidden;"><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0 tabular-nums" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-numeric-spacing: tabular-nums; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; flex-grow: 1; flex-shrink: 0; font-variant-numeric: var(--tw-ordinal) var(--tw-slashed-zero) var(--tw-numeric-figure) var(--tw-numeric-spacing) var(--tw-numeric-fraction);"></span><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; position: relative; width: calc(100% - 115px);"><div class="flex flex-grow flex-col gap-3 max-w-full" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-grow: 1; gap: 0.75rem; max-width: 100%;"><div class="min-h-[20px] flex flex-col items-start gap-3 overflow-x-auto whitespace-pre-wrap break-words" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-start; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; min-height: 20px; overflow-wrap: break-word; overflow-x: auto; white-space: pre-wrap;"><div class="empty:hidden" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;">solve the last equation for y</div></div></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 100%; --tw-translate-y: 0; align-self: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(172,172,190,var(--tw-text-opacity)); display: flex; gap: 0.25rem; justify-content: center; margin-top: 0px; padding-left: 0.5rem; position: absolute; right: 0px; top: 0px; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: visible;"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400 md:invisible md:group-hover:visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem; visibility: hidden;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div><div class="flex justify-between lg:block" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; justify-content: space-between;"></div></div></div></div></div><div class="group w-full text-token-text-primary border-b border-black/10 gizmo:border-0 dark:border-gray-900/50 gizmo:dark:border-0 bg-gray-50 gizmo:bg-transparent dark:bg-[#444654] gizmo:dark:bg-transparent" data-testid="conversation-turn-5" style="--avatar-color: #AB68FF; --tw-bg-opacity: 1; --tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-color: rgba(0, 0, 0, 0.1); border-image: initial; border-style: solid; border-width: 0px 0px 1px; box-sizing: border-box; color: var(--text-primary); font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 14px; width: 1172px;"><div class="p-4 justify-center text-base md:gap-6 md:py-6 m-auto" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-size: 1rem; gap: 1.5rem; justify-content: center; line-height: 1.5rem; margin: auto; padding: 1.5rem 1rem;"><div class="flex flex-1 gap-4 text-base mx-auto md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl }" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex: 1 1 0%; font-size: 1rem; gap: 1.5rem; line-height: 1.5rem; margin-left: auto; margin-right: auto; max-width: 48rem;"><div class="flex-shrink-0 flex flex-col relative items-end" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-end; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-shrink: 0; position: relative;"><div style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><div class="relative p-1 rounded-sm h-9 w-9 text-white flex items-center justify-center" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; align-items: center; background-color: #ab68ff; border-radius: 0.125rem; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(255,255,255,var(--tw-text-opacity)); display: flex; height: 36px; justify-content: center; padding: 0.25rem; position: relative; width: 36px;"><svg class="icon-md" fill="none" height="41" role="img" viewbox="0 0 41 41" width="41" xmlns="http://www.w3.org/2000/svg"><text x="-9999" y="-9999">ChatGPT</text><path d="M37.5324 16.8707C37.9808 15.5241 38.1363 14.0974 37.9886 12.6859C37.8409 11.2744 37.3934 9.91076 36.676 8.68622C35.6126 6.83404 33.9882 5.3676 32.0373 4.4985C30.0864 3.62941 27.9098 3.40259 25.8215 3.85078C24.8796 2.7893 23.7219 1.94125 22.4257 1.36341C21.1295 0.785575 19.7249 0.491269 18.3058 0.500197C16.1708 0.495044 14.0893 1.16803 12.3614 2.42214C10.6335 3.67624 9.34853 5.44666 8.6917 7.47815C7.30085 7.76286 5.98686 8.3414 4.8377 9.17505C3.68854 10.0087 2.73073 11.0782 2.02839 12.312C0.956464 14.1591 0.498905 16.2988 0.721698 18.4228C0.944492 20.5467 1.83612 22.5449 3.268 24.1293C2.81966 25.4759 2.66413 26.9026 2.81182 28.3141C2.95951 29.7256 3.40701 31.0892 4.12437 32.3138C5.18791 34.1659 6.8123 35.6322 8.76321 36.5013C10.7141 37.3704 12.8907 37.5973 14.9789 37.1492C15.9208 38.2107 17.0786 39.0587 18.3747 39.6366C19.6709 40.2144 21.0755 40.5087 22.4946 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font-size: 0.75rem; gap: 0.25rem; justify-content: center; left: 0px; line-height: 1rem; margin-left: -1rem; position: absolute; top: 0.5rem; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: hidden;"><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0 tabular-nums" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-numeric-spacing: tabular-nums; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; flex-grow: 1; flex-shrink: 0; font-variant-numeric: var(--tw-ordinal) var(--tw-slashed-zero) var(--tw-numeric-figure) var(--tw-numeric-spacing) var(--tw-numeric-fraction);"></span><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; position: relative; width: calc(100% - 115px);"><div class="flex flex-grow flex-col gap-3 max-w-full" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-grow: 1; gap: 0.75rem; max-width: 100%;"><div class="min-h-[20px] flex flex-col items-start gap-3 overflow-x-auto whitespace-pre-wrap break-words" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-start; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; min-height: 20px; overflow-wrap: break-word; overflow-x: auto; white-space: pre-wrap;"><div class="markdown prose w-full break-words dark:prose-invert light" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-prose-body: #374151; --tw-prose-bold: #111827; --tw-prose-bullets: #d1d5db; --tw-prose-captions: #6b7280; --tw-prose-code: #111827; --tw-prose-counters: #6b7280; --tw-prose-headings: #111827; --tw-prose-hr: #e5e7eb; --tw-prose-invert-body: #d1d5db; --tw-prose-invert-bold: #fff; --tw-prose-invert-bullets: #4b5563; --tw-prose-invert-captions: #9ca3af; --tw-prose-invert-code: #fff; --tw-prose-invert-counters: #9ca3af; --tw-prose-invert-headings: #fff; --tw-prose-invert-hr: #374151; --tw-prose-invert-lead: #9ca3af; --tw-prose-invert-links: #fff; --tw-prose-invert-pre-bg: rgba(0,0,0,0.5); --tw-prose-invert-pre-code: #d1d5db; --tw-prose-invert-quote-borders: #374151; --tw-prose-invert-quotes: #f3f4f6; --tw-prose-invert-td-borders: #374151; --tw-prose-invert-th-borders: #4b5563; --tw-prose-lead: #4b5563; --tw-prose-links: #111827; --tw-prose-pre-bg: #1f2937; --tw-prose-pre-code: #e5e7eb; --tw-prose-quote-borders: #e5e7eb; --tw-prose-quotes: #111827; --tw-prose-td-borders: #e5e7eb; --tw-prose-th-borders: #d1d5db; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: var(--tw-prose-body); font-size: 1rem; line-height: 1.75; max-width: none; overflow-wrap: break-word; width: 653px;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px 0px 1.25em;">You want to solve the equation of the circle:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(x - 0.5)^2 + y^2 = (0.5)^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">for <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.625em; vertical-align: -0.1944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span></span></span></span></span>.</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Expanding the equation, we get:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">x^2 - x + 0.25 + y^2 = 0.25</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6667em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.7278em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Simplifying and solving for <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">y^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span>:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">y^2 = 0.25 - x^2 + x</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.7278em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.4306em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Now, taking the square root of both sides:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">y = \pm \sqrt{0.25 - x^2 + x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.625em; vertical-align: -0.1944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.04em; vertical-align: -0.1266em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">±</span><span class="mord sqrt" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t vlist-t2" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; margin-right: -2px; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.9134em; position: relative; vertical-align: bottom;"><span class="svg-align" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative; top: -3em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 3em; overflow: hidden; width: 0px;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; padding-left: 0.833em;"><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.7401em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -2.989em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span></span></span><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative; top: -2.8734em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 3em; overflow: hidden; width: 0px;"></span><span class="hide-tail" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.08em; min-width: 0.853em; overflow: hidden; position: relative; width: 128.266px;"><svg height="1.08em" preserveaspectratio="xMinYMin slice" viewbox="0 0 400000 1080" width="400em" xmlns="http://www.w3.org/2000/svg"><path d="M95,702
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--tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.1266em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative;"></span></span></span></span></span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px 0px;">This will give two solutions, corresponding to the upper and lower halves of the circle.</p></div></div></div><div class="flex justify-between lg:block" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; justify-content: space-between;"><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 100%; --tw-translate-y: 0; align-self: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(172,172,190,var(--tw-text-opacity)); display: flex; gap: 0.25rem; justify-content: center; margin-top: 0px; padding-left: 0.5rem; position: absolute; right: 0px; top: 0px; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: visible;"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; display: flex; font-family: inherit; font-size: 16px; font-weight: inherit; gap: 0.5rem; line-height: inherit; margin: 0px 0px 0px auto; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect height="4" rx="1" ry="1" width="8" x="8" y="2"></rect></svg></button><div class="flex gap-1" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; gap: 0.25rem;"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div></div><div class="group w-full text-token-text-primary border-b border-black/10 gizmo:border-0 dark:border-gray-900/50 gizmo:dark:border-0 gizmo:bg-transparent dark:bg-gray-800 gizmo:dark:bg-transparent" data-testid="conversation-turn-6" style="--avatar-color: #AB68FF; --tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-color: rgba(0, 0, 0, 0.1); border-image: initial; border-style: solid; border-width: 0px 0px 1px; box-sizing: border-box; color: var(--text-primary); font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 14px; width: 1172px;"><div class="p-4 justify-center text-base md:gap-6 md:py-6 m-auto" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-size: 1rem; gap: 1.5rem; justify-content: center; line-height: 1.5rem; margin: auto; padding: 1.5rem 1rem;"><div class="flex flex-1 gap-4 text-base mx-auto md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl }" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex: 1 1 0%; font-size: 1rem; gap: 1.5rem; line-height: 1.5rem; margin-left: auto; margin-right: auto; max-width: 48rem;"><div class="flex-shrink-0 flex flex-col relative items-end" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-end; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-shrink: 0; position: relative;"><div style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><div class="relative flex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; position: relative;"><img alt="User" class="rounded-sm" data-nimg="1" decoding="async" height="36" loading="lazy" src="https://chat.openai.com/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=96&q=75" srcset="/_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=48&q=75 1x, /_next/image?url=https%3A%2F%2Fs.gravatar.com%2Favatar%2F2afd93b9096c72bba4aa6921951c0095%3Fs%3D480%26r%3Dpg%26d%3Dhttps%253A%252F%252Fcdn.auth0.com%252Favatars%252Fmc.png&w=96&q=75 2x" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-radius: 0.125rem; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: transparent; display: block; height: auto; max-width: 100%; vertical-align: middle;" width="36" /></div></div><div class="text-xs flex items-center justify-center gap-1 absolute left-0 top-2 -ml-4 -translate-x-full invisible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: -100%; --tw-translate-y: 0; align-items: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; font-size: 0.75rem; gap: 0.25rem; justify-content: center; left: 0px; line-height: 1rem; margin-left: -1rem; position: absolute; top: 0.5rem; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: hidden;"><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0 tabular-nums" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-numeric-spacing: tabular-nums; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; flex-grow: 1; flex-shrink: 0; font-variant-numeric: var(--tw-ordinal) var(--tw-slashed-zero) var(--tw-numeric-figure) var(--tw-numeric-spacing) var(--tw-numeric-fraction);"></span><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; position: relative; width: calc(100% - 115px);"><div class="flex flex-grow flex-col gap-3 max-w-full" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-grow: 1; gap: 0.75rem; max-width: 100%;"><div class="min-h-[20px] flex flex-col items-start gap-3 overflow-x-auto whitespace-pre-wrap break-words" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-start; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; min-height: 20px; overflow-wrap: break-word; overflow-x: auto; white-space: pre-wrap;"><div class="empty:hidden" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;">are you sure about that last equation?</div></div></div><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 100%; --tw-translate-y: 0; align-self: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(172,172,190,var(--tw-text-opacity)); display: flex; gap: 0.25rem; justify-content: center; margin-top: 0px; padding-left: 0.5rem; position: absolute; right: 0px; top: 0px; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: visible;"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400 md:invisible md:group-hover:visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem; visibility: hidden;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M11 4H4a2 2 0 0 0-2 2v14a2 2 0 0 0 2 2h14a2 2 0 0 0 2-2v-7"></path><path d="M18.5 2.5a2.121 2.121 0 0 1 3 3L12 15l-4 1 1-4 9.5-9.5z"></path></svg></button></div><div class="flex justify-between lg:block" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; justify-content: space-between;"></div></div></div></div></div><div class="group w-full text-token-text-primary border-b border-black/10 gizmo:border-0 dark:border-gray-900/50 gizmo:dark:border-0 bg-gray-50 gizmo:bg-transparent dark:bg-[#444654] gizmo:dark:bg-transparent" data-testid="conversation-turn-7" style="--avatar-color: #AB68FF; --tw-bg-opacity: 1; --tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-color: rgba(0, 0, 0, 0.1); border-image: initial; border-style: solid; border-width: 0px 0px 1px; box-sizing: border-box; color: var(--text-primary); font-family: Söhne, ui-sans-serif, system-ui, -apple-system, "Segoe UI", Roboto, Ubuntu, Cantarell, "Noto Sans", sans-serif, "Helvetica Neue", Arial, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 14px; width: 1172px;"><div class="p-4 justify-center text-base md:gap-6 md:py-6 m-auto" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-size: 1rem; gap: 1.5rem; justify-content: center; line-height: 1.5rem; margin: auto; padding: 1.5rem 1rem;"><div class="flex flex-1 gap-4 text-base mx-auto md:gap-6 md:max-w-2xl lg:max-w-[38rem] xl:max-w-3xl }" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex: 1 1 0%; font-size: 1rem; gap: 1.5rem; line-height: 1.5rem; margin-left: auto; margin-right: auto; max-width: 48rem;"><div class="flex-shrink-0 flex flex-col relative items-end" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-end; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-shrink: 0; position: relative;"><div style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><div class="relative p-1 rounded-sm h-9 w-9 text-white flex items-center justify-center" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; align-items: center; background-color: #ab68ff; border-radius: 0.125rem; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(255,255,255,var(--tw-text-opacity)); display: flex; height: 36px; justify-content: center; padding: 0.25rem; 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solid rgb(217, 217, 227); box-sizing: border-box; display: flex; font-size: 0.75rem; gap: 0.25rem; justify-content: center; left: 0px; line-height: 1rem; margin-left: -1rem; position: absolute; top: 0.5rem; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: hidden;"><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="15 18 9 12 15 6"></polyline></svg></button><span class="flex-grow flex-shrink-0 tabular-nums" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-numeric-spacing: tabular-nums; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; flex-grow: 1; flex-shrink: 0; font-variant-numeric: var(--tw-ordinal) var(--tw-slashed-zero) var(--tw-numeric-figure) var(--tw-numeric-spacing) var(--tw-numeric-fraction);"></span><button class="dark:text-white disabled:text-gray-300 dark:disabled:text-gray-400" disabled="" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-style: solid; border-width: 0px; font-family: inherit; font-size: 12px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px;"><svg class="icon-xs" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><polyline points="9 18 15 12 9 6"></polyline></svg></button></div></div><div class="relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; position: relative; width: calc(100% - 115px);"><div class="flex flex-grow flex-col gap-3 max-w-full" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; flex-grow: 1; gap: 0.75rem; max-width: 100%;"><div class="min-h-[20px] flex flex-col items-start gap-3 overflow-x-auto whitespace-pre-wrap break-words" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; align-items: flex-start; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; flex-direction: column; gap: 0.75rem; min-height: 20px; overflow-wrap: break-word; overflow-x: auto; white-space: pre-wrap;"><div class="markdown prose w-full break-words dark:prose-invert light" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-prose-body: #374151; --tw-prose-bold: #111827; --tw-prose-bullets: #d1d5db; --tw-prose-captions: #6b7280; --tw-prose-code: #111827; --tw-prose-counters: #6b7280; --tw-prose-headings: #111827; --tw-prose-hr: #e5e7eb; --tw-prose-invert-body: #d1d5db; --tw-prose-invert-bold: #fff; --tw-prose-invert-bullets: #4b5563; --tw-prose-invert-captions: #9ca3af; --tw-prose-invert-code: #fff; --tw-prose-invert-counters: #9ca3af; --tw-prose-invert-headings: #fff; --tw-prose-invert-hr: #374151; --tw-prose-invert-lead: #9ca3af; --tw-prose-invert-links: #fff; --tw-prose-invert-pre-bg: rgba(0,0,0,0.5); --tw-prose-invert-pre-code: #d1d5db; --tw-prose-invert-quote-borders: #374151; --tw-prose-invert-quotes: #f3f4f6; --tw-prose-invert-td-borders: #374151; --tw-prose-invert-th-borders: #4b5563; --tw-prose-lead: #4b5563; --tw-prose-links: #111827; --tw-prose-pre-bg: #1f2937; --tw-prose-pre-code: #e5e7eb; --tw-prose-quote-borders: #e5e7eb; --tw-prose-quotes: #111827; --tw-prose-td-borders: #e5e7eb; --tw-prose-th-borders: #d1d5db; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: var(--tw-prose-body); font-size: 1rem; line-height: 1.75; max-width: none; overflow-wrap: break-word; width: 653px;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px 0px 1.25em;">Let's reconsider the solution:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Starting from the equation:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">(x - 0.5)^2 + y^2 = (0.5)^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.5</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">x^2 - x + 0.25 + y^2 = 0.25</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6667em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.7278em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Subtracting <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">0.25</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0.25</span></span></span></span></span> from both sides:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">x^2 - x + y^2 = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6667em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Rearranging for <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">y^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span>:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">y^2 = x - x^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0085em; vertical-align: -0.1944em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6667em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;">Now, taking the square root of both sides:</p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">y = \pm \sqrt{x - x^2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.625em; vertical-align: -0.1944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: 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inline-block; height: 1.04em; vertical-align: -0.1266em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">±</span><span class="mord sqrt" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; 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--tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: 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transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: 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style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; 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--tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.1266em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative;"></span></span></span></span></span></span></span></span></span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 1.25em 0px 0px;">This is the correct expression for <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.625em; vertical-align: -0.1944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">y</span></span></span></span></span> in terms of <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 1.21em; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.4306em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span></span></span></span></span> for the circle. My apologies for the oversight in the previous response.</p><div><br /></div></div></div></div><div class="flex justify-between lg:block" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; justify-content: space-between;"><div class="text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-2 md:gap-3 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-text-opacity: 1; --tw-translate-x: 100%; --tw-translate-y: 0; align-self: center; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: rgba(172,172,190,var(--tw-text-opacity)); display: flex; gap: 0.25rem; justify-content: center; margin-top: 0px; padding-left: 0.5rem; position: absolute; right: 0px; top: 0px; transform: translate(var(--tw-translate-x),var(--tw-translate-y)) rotate(var(--tw-rotate)) skewX(var(--tw-skew-x)) skewY(var(--tw-skew-y)) scaleX(var(--tw-scale-x)) scaleY(var(--tw-scale-y)); visibility: visible;"><button class="flex ml-auto gap-2 rounded-md p-1 hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; display: flex; font-family: inherit; font-size: 16px; font-weight: inherit; gap: 0.5rem; line-height: inherit; margin: 0px 0px 0px auto; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M16 4h2a2 2 0 0 1 2 2v14a2 2 0 0 1-2 2H6a2 2 0 0 1-2-2V6a2 2 0 0 1 2-2h2"></path><rect height="4" rx="1" ry="1" width="8" x="8" y="2"></rect></svg></button><div class="flex gap-1" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; display: flex; gap: 0.25rem;"><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M14 9V5a3 3 0 0 0-3-3l-4 9v11h11.28a2 2 0 0 0 2-1.7l1.38-9a2 2 0 0 0-2-2.3zM7 22H4a2 2 0 0 1-2-2v-7a2 2 0 0 1 2-2h3"></path></svg></button><button class="p-1 rounded-md hover:bg-gray-100 hover:text-gray-700 dark:text-gray-400 dark:hover:bg-gray-700 dark:hover:text-gray-200 disabled:dark:hover:text-gray-400" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; appearance: button; background-image: none; border-color: rgb(217, 217, 227); border-radius: 0.375rem; border-style: solid; border-width: 0px; cursor: pointer; font-family: inherit; font-size: 16px; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0.25rem;"><svg class="icon-sm" fill="none" height="1em" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" stroke="currentColor" viewbox="0 0 24 24" width="1em" xmlns="http://www.w3.org/2000/svg"><path d="M10 15v4a3 3 0 0 0 3 3l4-9V2H5.72a2 2 0 0 0-2 1.7l-1.38 9a2 2 0 0 0 2 2.3zm7-13h2.67A2.31 2.31 0 0 1 22 4v7a2.31 2.31 0 0 1-2.33 2H17"></path></svg></button></div></div></div></div></div></div></div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-44912650856605010802023-09-20T16:13:00.009-07:002023-09-21T17:22:54.469-07:00More Cellular Automata<div style="text-align: justify;">I now have code that will use either 3-bit or 5-bit rules.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Rules can be applied from one to three layers. That is, Layer 1 would apply Rule 1 to the initial generation. Layer 2 would take the output of Layer 1 and applies Rule 2 to it. Layer 3 takes the output of Layer 2 and applies Rule 3 to it. The output of this would be used in the next generation where each Rule would again be applied, etc.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The code is flexible enough to allow me to choose 1, 2, or 3 layers.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The code can also be set up to keep one or more Rule constant in a layer, and loop through the range of Rules in another layer for a given Rule size (3-bit --> 256 total rules, 5-bit --> 4294967296 total rules).</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I run this for 2048 generations, and I make the window 2048 pixels wide (so later on my FFT is happy). I therefore end up with 'images' that are 2048x2048 pixels in size. I can then make some statistical measurements on these images, which means I compute the mean and standard deviation for each image produced.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">For instance, in the 'simplest' case, I can apply every 3-bit rule to a starting generation that's a single pixel at location 1023 in full 2048 pixel generation. If I do this, I'll get 256 'images' each corresponding to the application of that rule for 2048 generations.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The stats look like this:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgwlbB9zG6T2Srdzd71GZrjjXOSUO0eSvKxA1xJbqpc8Sr0NQv26-zbCLbscswVstl6rfjAbOiVG6pZS6xOrYzxf0-WlDR8z6kom_Gi4ABEe39j9wlxoCkXb-jm9mF-wh_iPxUDaBnDm90Sm9z0JwBJOLbCCCu6D9Cr-NbQx2x99U20oN5D4J20jLAkuQAg" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="1356" height="160" src="https://blogger.googleusercontent.com/img/a/AVvXsEgwlbB9zG6T2Srdzd71GZrjjXOSUO0eSvKxA1xJbqpc8Sr0NQv26-zbCLbscswVstl6rfjAbOiVG6pZS6xOrYzxf0-WlDR8z6kom_Gi4ABEe39j9wlxoCkXb-jm9mF-wh_iPxUDaBnDm90Sm9z0JwBJOLbCCCu6D9Cr-NbQx2x99U20oN5D4J20jLAkuQAg=w400-h160" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div style="text-align: justify;">The x-axis is the Rule number. The y-axis is the mean value for each 'image' produced by the Rule specified. The error bars represent the 1-sigma standard deviation. Purple lines are 0, 0.5, and 1. The red line indicates Rule 128.</div></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Some observations. A pixel can either have a value of zero or one. So an image with a mean of zero with a small standard deviation is mostly black. An image with a mean of one with a small standard deviation is mostly white. An image with a mean of 0.5 and a standard deviation of +/- 0.5 would be almost completely gray (or some combination of equal white and black).</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The mean and standard deviation have an interesting relationship (x-axis is the mean, y-axis is the stddev):</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgq_3nHtUI66xkGv0lq-d2k3lYTMw1guL2ja3O5VrQYVIYJsN9uZFdP1XzxYr27o0qIZB_eN_mrIHABWqQxyqmkNsC8MicJW4qx0n1JXVcNbiX873k1FwYpd9vR-YHI02Q7U1vih8yFVKGiov_5D2x7vO3861zGHkUFG3v8IcG3ryqifT-0He0MAe9Mqk73" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="1356" height="160" src="https://blogger.googleusercontent.com/img/a/AVvXsEgq_3nHtUI66xkGv0lq-d2k3lYTMw1guL2ja3O5VrQYVIYJsN9uZFdP1XzxYr27o0qIZB_eN_mrIHABWqQxyqmkNsC8MicJW4qx0n1JXVcNbiX873k1FwYpd9vR-YHI02Q7U1vih8yFVKGiov_5D2x7vO3861zGHkUFG3v8IcG3ryqifT-0He0MAe9Mqk73=w400-h160" width="400" /></a></div><br /><br /></div><div style="text-align: justify;">The 'intersting' stuff looks like to be then the mean is around 0.25 or 0.75, and the standard deviation isn't small (near zero) or high (+/- 0.5).</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Here's stats on the famous Rule 30 and it's immediate neighbors. X-axis is the Rule number, y-axis is the mean, error bars are the 1-sigma standard deviation. Rule 30 does have an unusual characteristic in that it's the last Rule (for a little while) where the 'images' are either all black or all gray.</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhhpdUAj2iZgxaplXcs-Y3WTw0nIexYjY8i3mq54XJiWzByn-3PRtZYhQpIhNXnHlzLj3hZ5s5hOVX3usz7FREKM-7FMe4D6Qsq462Jkldu2Ia1cywoS2O6rdN-I-vXkcaYNPqEQgGwktMrBObuzqvchj_qDNL-NTHfQorAUOCkzSVHyLyFLQ73pXblCLpt" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="1356" height="160" src="https://blogger.googleusercontent.com/img/a/AVvXsEhhpdUAj2iZgxaplXcs-Y3WTw0nIexYjY8i3mq54XJiWzByn-3PRtZYhQpIhNXnHlzLj3hZ5s5hOVX3usz7FREKM-7FMe4D6Qsq462Jkldu2Ia1cywoS2O6rdN-I-vXkcaYNPqEQgGwktMrBObuzqvchj_qDNL-NTHfQorAUOCkzSVHyLyFLQ73pXblCLpt=w400-h160" width="400" /></a></div><br /><br /></div><div><br /></div><div>Here's the 'image' generated of 2048 generations of Rule 90:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgJJhnF8gxCbI9EJ9YxJoUoycYbfv6JiXgK0zhWb0zRTtjOIrkIbxsc-yE0l5rWU28Brt2ddBmLme5nXjFoQhqCh32SNw-XsVjBy44QCM7jsd7Qt9CDjlHLdCZ1DpwGzE9L0GFDgc9KsEMPUPVIXcxnGxWitH4aUtpn0-mWM2hFzbVfAiESBmbxHCZmYQlt" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="511" data-original-width="511" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEgJJhnF8gxCbI9EJ9YxJoUoycYbfv6JiXgK0zhWb0zRTtjOIrkIbxsc-yE0l5rWU28Brt2ddBmLme5nXjFoQhqCh32SNw-XsVjBy44QCM7jsd7Qt9CDjlHLdCZ1DpwGzE9L0GFDgc9KsEMPUPVIXcxnGxWitH4aUtpn0-mWM2hFzbVfAiESBmbxHCZmYQlt=w400-h400" width="400" /></a></div><br /><br /></div><div>Rule 92, which looks very similar <i>statistically</i> to Rule 90, looks very different:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjflQx_eT89368meEOjk1lIGO2tmbnsej4RkNtgJuvcT3d1A2-pm9wHe1_iOBmjxpv6Rm6TAuQSZvdaHhc_fz6IKYWIN0iQTpIHKOYjr9tf8v6Coky7i4DGdiTVqM3deEE0vO6kSb9kBhXL0iW5I0ffD6K_vwKl-574SOFLuWl7h4bKZyCc0YGhvgxl303m" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="510" data-original-width="510" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEjflQx_eT89368meEOjk1lIGO2tmbnsej4RkNtgJuvcT3d1A2-pm9wHe1_iOBmjxpv6Rm6TAuQSZvdaHhc_fz6IKYWIN0iQTpIHKOYjr9tf8v6Coky7i4DGdiTVqM3deEE0vO6kSb9kBhXL0iW5I0ffD6K_vwKl-574SOFLuWl7h4bKZyCc0YGhvgxl303m=w400-h400" width="400" /></a></div><br /></div><div><br /></div><div>So this tells me that stats like these have limited use.</div><div><br /></div><div>My code will also move this into frequency space. Here's the FFT of Rule 90:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhXq9pam8rguS7QK0HIsmYRtYLV6cYyWPO4e6bYmOJujauZfgh9WV4aVmyfBbZCgIR_7igF6ifTWpvQgRWRGuCuuMQpCv8rHm0dY-t54CHyIvkBJYEi8JM8SnrscfTzXIPDOKbO5zx0PWoer6JRFYw-kJIT5dPTL8jndzI36587vSOWoYZKcLPV8JAUCr32" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="512" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEhXq9pam8rguS7QK0HIsmYRtYLV6cYyWPO4e6bYmOJujauZfgh9WV4aVmyfBbZCgIR_7igF6ifTWpvQgRWRGuCuuMQpCv8rHm0dY-t54CHyIvkBJYEi8JM8SnrscfTzXIPDOKbO5zx0PWoer6JRFYw-kJIT5dPTL8jndzI36587vSOWoYZKcLPV8JAUCr32=w400-h400" width="400" /></a></div><br /></div><div><br /></div><div style="text-align: justify;">I also have code that will take this data and convert it to audio.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">But of course now I can apply different layers --> apply multiple rules to a single generation. This is where things seem to get interesting.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I'll fix Rule 90 for the first layer, and then run through the 256, 3-bit rules in layer 2. Here are the stats:</div><div style="text-align: justify;"><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhS-Ua8wnKjCTmMQX7s-DhWJp-GOc5vXGkNIv0ILsm3EifMDfQurFoyAU9-50Kt1kRO6764TuQf5mBlawz8YdpKhAVhjZSQs4ArkQkAOs_PsWIIiRX0wHAQFDkPs7QfiKcSQkwuJ3iKhXAnoL28a_ufaxqTexIJzlIwGMpVaY9Hzbl81uGoAobYwtIVkpkL" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="1361" height="160" src="https://blogger.googleusercontent.com/img/a/AVvXsEhS-Ua8wnKjCTmMQX7s-DhWJp-GOc5vXGkNIv0ILsm3EifMDfQurFoyAU9-50Kt1kRO6764TuQf5mBlawz8YdpKhAVhjZSQs4ArkQkAOs_PsWIIiRX0wHAQFDkPs7QfiKcSQkwuJ3iKhXAnoL28a_ufaxqTexIJzlIwGMpVaY9Hzbl81uGoAobYwtIVkpkL=w400-h160" width="400" /></a></div><br /><br /></div><div>Compare it to this plot of a fixed Rule 1 in the first layer, and the the 256 rules for layer 2:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgaV5tzpur8Zw_c6s94O7oX--cPFlrcO8hQPDiLF6-Q1u2ZkCE8LCYpvRl0zh79m6DYtggEaU_iwoz8OCEBJdHnc9H8BshBLFKhPtzZLv8FzAdfnWQI57xz_ibR3hx8JGmeg8JTFP_y1DawBWMj7fElBOMh7AIVD8iPqy3CRo_3tAWgQEUJO1joA6-kxkgD" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="1377" height="158" src="https://blogger.googleusercontent.com/img/a/AVvXsEgaV5tzpur8Zw_c6s94O7oX--cPFlrcO8hQPDiLF6-Q1u2ZkCE8LCYpvRl0zh79m6DYtggEaU_iwoz8OCEBJdHnc9H8BshBLFKhPtzZLv8FzAdfnWQI57xz_ibR3hx8JGmeg8JTFP_y1DawBWMj7fElBOMh7AIVD8iPqy3CRo_3tAWgQEUJO1joA6-kxkgD=w400-h158" width="400" /></a></div></div><div><br /></div><div>Curious and interesting...</div><div><br /></div><div>To come to a logical conclusion, I've fixed layer 1 to be Rule 1, and layer 2 to be Rule 49, and then ran through all the Rules for layer 3. I get these stats:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjUtJKPwqWOI-YuBY0RikB8zDipoE3VNM1KMasE58m4z8tf_xAa_xt5aV0hNTK75vv2t22gChn_RRIWgGpXVYpc6mPXzd1RmbyN3jxZDl0EEsbT0ENxaVXXTwiZuqWpq4ABboMJStCRtDPHBYioXl9c2Pw4mbapx7TvINFXJosyv8vU4QJshHcxZyV4WDFz" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="1405" height="155" src="https://blogger.googleusercontent.com/img/a/AVvXsEjUtJKPwqWOI-YuBY0RikB8zDipoE3VNM1KMasE58m4z8tf_xAa_xt5aV0hNTK75vv2t22gChn_RRIWgGpXVYpc6mPXzd1RmbyN3jxZDl0EEsbT0ENxaVXXTwiZuqWpq4ABboMJStCRtDPHBYioXl9c2Pw4mbapx7TvINFXJosyv8vU4QJshHcxZyV4WDFz=w400-h155" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br />Here's the result of Rule 1 x Rule 49 x Rule 66:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhkilqalZDla_eP65qn2FtPKMMNUnZwkO5vC3eWYAR38zw22yXCmvSo6A33Rh6vWisx16mxTqpcLWw1m5ZGP-dKTot6gBdjie6Z5dqQwMjG-lmNVobNFSqcycP7aYQTLqatQ3uV4ZNQmIpN-BLvPwrBr5B57k9Z2Q7OmhoUwf1qQvfqdzKrUesuWU6t2PgK" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="514" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEhkilqalZDla_eP65qn2FtPKMMNUnZwkO5vC3eWYAR38zw22yXCmvSo6A33Rh6vWisx16mxTqpcLWw1m5ZGP-dKTot6gBdjie6Z5dqQwMjG-lmNVobNFSqcycP7aYQTLqatQ3uV4ZNQmIpN-BLvPwrBr5B57k9Z2Q7OmhoUwf1qQvfqdzKrUesuWU6t2PgK=w398-h400" width="398" /></a></div><br />and the power spectrum:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEh3zXbOmxEXmm6-sUI53gUiuKCwlrmwvbctmCifHJJIisccM2MSen1y8nuVKb_lryT1GWYIZH6XL0f9LC397GcONZTgxtfxsQUi-tt4CYOptYf9Xp2kOOleviOjM0H4yUVGm7zA8XmzHghA8zaid0fDvhg9Ehw6JDVAGRitjcsHoqisYUwjAr3Iktgm8r4-" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="513" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEh3zXbOmxEXmm6-sUI53gUiuKCwlrmwvbctmCifHJJIisccM2MSen1y8nuVKb_lryT1GWYIZH6XL0f9LC397GcONZTgxtfxsQUi-tt4CYOptYf9Xp2kOOleviOjM0H4yUVGm7zA8XmzHghA8zaid0fDvhg9Ehw6JDVAGRitjcsHoqisYUwjAr3Iktgm8r4-=w400-h400" width="400" /></a></div><br />detail of a section of the power spectrum:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi_C9MFldN23xaNYoiYcWJtVgIb6j9vi9VCrpcNExPAkXafuQhg26UgqINr2UkrI4sZMwHty97qgbMkAdmxArx6n6e68rAO3frkWVGWTzwNRZBtEBnBYlNcZDzOy3pjaS8GQPMRhe5neUSZ0noRA-X22M5LLQ8Wl9FTXR1C6QaYmdy9c99AIu9UiGVrLBks" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="677" data-original-width="854" height="317" src="https://blogger.googleusercontent.com/img/a/AVvXsEi_C9MFldN23xaNYoiYcWJtVgIb6j9vi9VCrpcNExPAkXafuQhg26UgqINr2UkrI4sZMwHty97qgbMkAdmxArx6n6e68rAO3frkWVGWTzwNRZBtEBnBYlNcZDzOy3pjaS8GQPMRhe5neUSZ0noRA-X22M5LLQ8Wl9FTXR1C6QaYmdy9c99AIu9UiGVrLBks=w400-h317" width="400" /></a></div><br />If I was able to post the sound to this blog, I would. I'll put it somewhere and make a link to it.</div><div><br /></div><div>An interesting mystery is the following: Given the stats for a set of Rules:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhUqBmKx63HneYmnIKPHOt8ZkANPtFexI3ZOAQJYutJrzKigMdVlzrqHaEDNlJ6VpgSr9baOaJdPYYwTCZ2e2LUVtPXbx5h3b4wQiCUT0o1_jqwwyGDJzRtLsEfqJcOzP33cIaAGtTuppLHMNS887nYigBqot5Y77RVK0bcre99dRB61uG-P8iv5I0upEyD" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="474" data-original-width="1405" height="135" src="https://blogger.googleusercontent.com/img/a/AVvXsEhUqBmKx63HneYmnIKPHOt8ZkANPtFexI3ZOAQJYutJrzKigMdVlzrqHaEDNlJ6VpgSr9baOaJdPYYwTCZ2e2LUVtPXbx5h3b4wQiCUT0o1_jqwwyGDJzRtLsEfqJcOzP33cIaAGtTuppLHMNS887nYigBqot5Y77RVK0bcre99dRB61uG-P8iv5I0upEyD=w400-h135" width="400" /></a></div><br /><br /></div><div>If I plot mean value versus standard deviation, I get an unexpected relationship:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgjHncYEBn2livLcO4TlZulsI70AvOdWPi8kP9UagwNz9s7J222BYfFPfTu76dDGPct4pa3IuSEeygyCJGJXYWACc3JEeTUegTv4KOX5KitUHdylMQq8SiBlvD8p5Wi-hXoJ8pkuvBlu4RlVPY4owrYbITgOMBLjXiNUGFrOGHjUZ-Cn3qOGyq8odubeXv_" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="474" data-original-width="734" height="259" src="https://blogger.googleusercontent.com/img/a/AVvXsEgjHncYEBn2livLcO4TlZulsI70AvOdWPi8kP9UagwNz9s7J222BYfFPfTu76dDGPct4pa3IuSEeygyCJGJXYWACc3JEeTUegTv4KOX5KitUHdylMQq8SiBlvD8p5Wi-hXoJ8pkuvBlu4RlVPY4owrYbITgOMBLjXiNUGFrOGHjUZ-Cn3qOGyq8odubeXv_=w400-h259" width="400" /></a></div><br /><br /></div><div>... seem to follow a circular relationship, of the form:</div><div><br /></div><div><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; background-color: #f7f7f8; border: 0px solid currentcolor; box-sizing: border-box; color: #374151; display: inline-block; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 19.36px; position: relative; white-space: nowrap; width: min-content;"><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;"><span> </span><span> </span><span> </span><span> </span><span> </span>y</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; background-color: #f7f7f8; border: 0px solid currentcolor; box-sizing: border-box; color: #374151; display: inline-block; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 19.36px; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6667em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; background-color: #f7f7f8; border: 0px solid currentcolor; box-sizing: border-box; color: #374151; display: inline-block; font-family: KaTeX_Main, "Times New Roman", serif; font-size: 19.36px; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.8141em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">x</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,0.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">2</span></span></span></span></span></span></span></span></span></div><div><br /></div><div>I just checked, and the match is identical. So this means that every standard deviation is computable given every mean value from zero to one.</div><div><br /></div><div>Putting this into words:</div><div><br /></div><div>stddev = sqrt (mean - mean^2)</div><div><br /></div><div>not sure what to make of that...</div><div><br /></div><div>Here's what I get if I keep Rule 90 static and have the next layer loop through the 256 possible Rules. I get these results:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhR7LN48wA0xysjxyTk6TtmjWOEUljNv1OLg0cUThrOJ0owcOgaboYXAXyYPxbQdMAT8tCUm52kHq-Gidh95he2PRb9d2dFibWP--_dEOWSf-9jq8Gzq0h6hfEccxg8-zI37TaM90lX24XXc0mJWIzImUUMeUk3HT8N61iH6NFnPsctM6T4n0sd6LiEJdVJ" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="1289" height="169" src="https://blogger.googleusercontent.com/img/a/AVvXsEhR7LN48wA0xysjxyTk6TtmjWOEUljNv1OLg0cUThrOJ0owcOgaboYXAXyYPxbQdMAT8tCUm52kHq-Gidh95he2PRb9d2dFibWP--_dEOWSf-9jq8Gzq0h6hfEccxg8-zI37TaM90lX24XXc0mJWIzImUUMeUk3HT8N61iH6NFnPsctM6T4n0sd6LiEJdVJ=w400-h169" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiFqvD6g_itqxaz4o7VopY-vS9m3ooteDRu4o6AXeQ00CQLjbEDk1S-Wf3B-D5F-HDkqeqraMvPRCJu6yihhT9T6BAw_WvVnW5pF-Llz_qc-ZukYSeJ534P1mr2a7rlqEan5XRyWmYoaTgKpAJ5dyF2SXwthAnuHt_tDaiGelYDo-zEX_YFjFzqU1ee7L6y" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="885" height="245" src="https://blogger.googleusercontent.com/img/a/AVvXsEiFqvD6g_itqxaz4o7VopY-vS9m3ooteDRu4o6AXeQ00CQLjbEDk1S-Wf3B-D5F-HDkqeqraMvPRCJu6yihhT9T6BAw_WvVnW5pF-Llz_qc-ZukYSeJ534P1mr2a7rlqEan5XRyWmYoaTgKpAJ5dyF2SXwthAnuHt_tDaiGelYDo-zEX_YFjFzqU1ee7L6y=w400-h245" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Gotta understand that better!</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">No idea where I'm going with any of this...</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">---</div><div class="separator" style="clear: both; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgRDQPXZieTFqoORIRyJcjdL961yVj36qnuXID1pkgUiIrhHKjN_XGD2YQOJuRaUcdA4_3rxAKfAc_K6dS2FaWmEN7s60_cTa95Jak-do_EZzZaJG7GZUp8FMTFDpUMzNDRthgEerx7Oc8-2HxsRNBQ2_XFCzvUnJnqtkFWjlsjEcCD4NnP8P4EckB-Gfjp" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="886" height="245" src="https://blogger.googleusercontent.com/img/a/AVvXsEgRDQPXZieTFqoORIRyJcjdL961yVj36qnuXID1pkgUiIrhHKjN_XGD2YQOJuRaUcdA4_3rxAKfAc_K6dS2FaWmEN7s60_cTa95Jak-do_EZzZaJG7GZUp8FMTFDpUMzNDRthgEerx7Oc8-2HxsRNBQ2_XFCzvUnJnqtkFWjlsjEcCD4NnP8P4EckB-Gfjp=w400-h245" width="400" /></a></div><br /><div style="text-align: justify;">These points are the same as above, but now coded for which rule is being applied. Zero (black) to 255 (white).</div><br /></div>Obviously this is telling me that there is structure in the data itself. But what, and why? Maybe something to do with the fact that it's all 1's and 0's???? Gotta look into Binomial Distributions -- I think that's the ticket to explaining and understanding this.<br /></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">A rabbit hole, for sure...</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Here's yet another way to look at the data -- connecting all the dots in the path from Rule 0 to Rule 255:</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiw_QAQZwV9TkBvdWu06VzbqKU7MEtmtESLPwxHjtzrCiMv-9id8RHX3xT3tBzFhr5v2NNHUd8E98v7MRXmSLma6fh-7ppU8hi3ISkgoQ6BwhKLGtv0nJhNRmbi2wyU9EVZjEV_E13erh1blxMQ3OY1lFxOhATAZXn4X87EcEo6BtpJo-r8tFIum_JgCP70" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="885" height="245" src="https://blogger.googleusercontent.com/img/a/AVvXsEiw_QAQZwV9TkBvdWu06VzbqKU7MEtmtESLPwxHjtzrCiMv-9id8RHX3xT3tBzFhr5v2NNHUd8E98v7MRXmSLma6fh-7ppU8hi3ISkgoQ6BwhKLGtv0nJhNRmbi2wyU9EVZjEV_E13erh1blxMQ3OY1lFxOhATAZXn4X87EcEo6BtpJo-r8tFIum_JgCP70=w400-h245" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;">Layer 1 = Rule 90, Layer 2 = Rules 0:255</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjOiq2Cdm3MqV5V1b2zaPdd24h2jByGeZh8JlynmnWE-6WEqUC1oNPsopAhlXlxe2VeAMNCVGqXVckT9k2WtlqJUDU_NO4Uw0kAMySYksBOewFueOZXuwEVa7MrLDGdsAxGk-sVv__lHl1a92l8hk70zjLOwfxHrjRoS3nPP5n9f2hY98CyJCatNNIXBKe5" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="885" height="245" src="https://blogger.googleusercontent.com/img/a/AVvXsEjOiq2Cdm3MqV5V1b2zaPdd24h2jByGeZh8JlynmnWE-6WEqUC1oNPsopAhlXlxe2VeAMNCVGqXVckT9k2WtlqJUDU_NO4Uw0kAMySYksBOewFueOZXuwEVa7MrLDGdsAxGk-sVv__lHl1a92l8hk70zjLOwfxHrjRoS3nPP5n9f2hY98CyJCatNNIXBKe5=w400-h245" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;">Layer 1 = Rules 0:255</div></div></div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-35090106185808749042023-09-06T14:48:00.003-07:002023-09-06T14:48:59.654-07:00Cellular Automata ExplosionI've written a 5-bit version of the cellular automata (CA) code. So instead of looking at the value of the current and the 2 adjacent pixels (1 left and 1 right of the current position), I'm looking at the current and 4 adjacent pixels (2 left and 2 right). This mean that the number of Rules for 1 layer is 2^32.<div><br /></div><div>The code also allows for up to three CA layers. The output of one layer is the input to the next:</div><div><br /></div><div>Random Input -> Rule1 -> Rule2 -> Rule3 -> output</div><div><br /></div><div>This is a portion of the output when Rule1 = Rule2 = Rule3 = 789263618</div><div><br /></div><div>In binary, 789263618 = 0x01000000101011001101000011110100</div><div><br /></div><div>So the same Rule applied three times in a row. I get this:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg9h7NuDmSzJ2_ZhIexkoi-TdZUg6okmiIUzc1OnLambXMTs4OpaQCgmQCOWcTNFgMLi4s8ect5yE14n85A0emniBcE2c2OFE3ItplaXpeho5pB-QOevQH2wbeBxWODhGZtUHq7-AreLfEzFsIcjsCUYenkuL8s508qNfM-huZRSs1r9JaptUS6pmQ0cCiU" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="680" data-original-width="853" height="319" src="https://blogger.googleusercontent.com/img/a/AVvXsEg9h7NuDmSzJ2_ZhIexkoi-TdZUg6okmiIUzc1OnLambXMTs4OpaQCgmQCOWcTNFgMLi4s8ect5yE14n85A0emniBcE2c2OFE3ItplaXpeho5pB-QOevQH2wbeBxWODhGZtUHq7-AreLfEzFsIcjsCUYenkuL8s508qNfM-huZRSs1r9JaptUS6pmQ0cCiU=w400-h319" width="400" /></a></div><br /><br /></div><div>That's interesting, I suppose. I'm sure PhD dissertations can be written about it. But for me, the fourier transform is much more interesting. Here's the FFT of the portion shown above:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhYEyzkZG7UfVRqE2wNeXOcrQbcZpCg3wjv7Leqw1hVNPI-JU0CeQlnysuoHECWIjHgiYQ3_ZhHiCjfEj05q5P9rZBeIoiR6hmMhPD-diov6runrbvYhIBFyqrPObEiWz9hpYt9njGyTHbd1_hnxERumGZW_8Uo6wm7bhuZcqQfkZKvXS0vPIcF1ItQ7aNl" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="676" data-original-width="851" height="318" src="https://blogger.googleusercontent.com/img/a/AVvXsEhYEyzkZG7UfVRqE2wNeXOcrQbcZpCg3wjv7Leqw1hVNPI-JU0CeQlnysuoHECWIjHgiYQ3_ZhHiCjfEj05q5P9rZBeIoiR6hmMhPD-diov6runrbvYhIBFyqrPObEiWz9hpYt9njGyTHbd1_hnxERumGZW_8Uo6wm7bhuZcqQfkZKvXS0vPIcF1ItQ7aNl=w400-h318" width="400" /></a></div><br /><br /></div><div>And one of the first things I think of is -- how to turn this into sound. That'll require diving into inverse-FFTs. Fun stuff.</div><div><br /></div><div>In any case, maybe there are more 'interesting' CA recipes than I thought. I'm finding them almost everywhere.</div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-12959824193619359802023-09-05T21:29:00.000-07:002023-09-05T21:29:38.945-07:00Cellular AutomataI've been interested in CA for many many years, and finally today I wrote my first code that implements the making of 'images' based on one (or possibly several at once) of 256 rules.<div><br /></div><div>...</div><div><br /></div><div>Well, before I knew it I've gone off the deep end with this stuff. Making these things that show them 'evolve' is pretty cool, but it looks like there are only a few rules that make anything 'interesting'. From out of nowhere I go the idea of combining rules. So take the starting input, apply Rule 1, then use that as input to Rule 2, for input to Rule 3, etc. I decided to stop at combining three Rules. This appears to be very very very different than the rule combination on the wolfram website, where it's taking the final results of two different rules and applying xor, or, & and to the images to produce a result. These results are boring.</div><div><br /></div><div>The output of one Rule being the input to the next Rule is something I can't find any info on, which means I just haven't found anything yet (likely), or I've come up with something fairly novel.</div><div><br /></div><div>Here's what combining Rules 86 and 112 look like:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg_4hmMGbfuYANtZOYsFaSbEvkP03SAU0bp9wUAGQCcM5I4GlTu1tCucXRX3JU7uGVXOeNkLxgXY3lbWd8_s9OLEYDPBLObgrEjzOeeX9l601GwtPdQcSrj8o324-H5BHFqUmQaKSGipNJ2icbkqrkp4B9yUsXn8A8VaFJHvntAOG83FZIiuB-gXWhXyqp7" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="780" data-original-width="400" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEg_4hmMGbfuYANtZOYsFaSbEvkP03SAU0bp9wUAGQCcM5I4GlTu1tCucXRX3JU7uGVXOeNkLxgXY3lbWd8_s9OLEYDPBLObgrEjzOeeX9l601GwtPdQcSrj8o324-H5BHFqUmQaKSGipNJ2icbkqrkp4B9yUsXn8A8VaFJHvntAOG83FZIiuB-gXWhXyqp7=w205-h400" width="205" /></a></div><br /><br /></div><div>The patterning here is subtle, but if I look at the power spectrum of each row I see this spectacular thing:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEimdhrO1wDUm0mTqPiAtYVWce4AlVv2WVDOy8xhF3Scz6uSFvfBEo3U8JlGzRvuGYxIcCf36EcLDaMqoOPI-ISJunLHo9c4BFqfCb4Td7CV7z4yojaBKpGDu7M3W8BrEJdrV970EJ1eNhaY02PfuEW67Zrg91izTDDwks8hYnMpvkoSIFWfDh3IXrZbHb2n" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="781" data-original-width="400" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEimdhrO1wDUm0mTqPiAtYVWce4AlVv2WVDOy8xhF3Scz6uSFvfBEo3U8JlGzRvuGYxIcCf36EcLDaMqoOPI-ISJunLHo9c4BFqfCb4Td7CV7z4yojaBKpGDu7M3W8BrEJdrV970EJ1eNhaY02PfuEW67Zrg91izTDDwks8hYnMpvkoSIFWfDh3IXrZbHb2n=w205-h400" width="205" /></a></div><br /><br /></div><div>Here's the same thing with different starting points:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhjsQ_SaHE-fTQZgdbdTyd_GLCL05vgRoMJ_XcoBnm1MUbxxxUn1qyYvKOhcco6qEK4kS3b1RScqYiK7dgjPHcQBTOVeqNCunBx7ey0XV2Bapp4wrW4UBt57mnjM0q3k9hxkz6NVFBjiLDWcb2FDfdwxHZJvhaCjt61wv1UheAbhY1baob3jwMJW8S-Wobg" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="780" data-original-width="400" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEhjsQ_SaHE-fTQZgdbdTyd_GLCL05vgRoMJ_XcoBnm1MUbxxxUn1qyYvKOhcco6qEK4kS3b1RScqYiK7dgjPHcQBTOVeqNCunBx7ey0XV2Bapp4wrW4UBt57mnjM0q3k9hxkz6NVFBjiLDWcb2FDfdwxHZJvhaCjt61wv1UheAbhY1baob3jwMJW8S-Wobg=w205-h400" width="205" /></a></div><br /><br /></div><div>and the power spectrum:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjetTZvtfXaK6ijPDHE8YYQINAS42IRl9P2z4nJaEaLFW-2s-tdPdTcYSr2pUW096RVdN1zYA8jGuZR2UIa3g-daHOcqJUGRtx7xxlgJJOIQ-zXWF2t60BSnTGsd-31B883_rbvtWo6XhHMwdMB7NUpWjSvaOS60ON_4guYlSnH4sswHq0aETOI6IOemN48" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="781" data-original-width="400" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEjetTZvtfXaK6ijPDHE8YYQINAS42IRl9P2z4nJaEaLFW-2s-tdPdTcYSr2pUW096RVdN1zYA8jGuZR2UIa3g-daHOcqJUGRtx7xxlgJJOIQ-zXWF2t60BSnTGsd-31B883_rbvtWo6XhHMwdMB7NUpWjSvaOS60ON_4guYlSnH4sswHq0aETOI6IOemN48=w205-h400" width="205" /></a></div><br /><br /></div><div>details of the PS:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjodNq5_5wl409M8M-p6JSIPHPG95HV3q11F1Y85rCitk7yW_4c5jxfp3VNX3nnxH6sDyYlgUVrqIA_wDRDsjsx1taBr2BjXx9JZMxdkKHZIZCGxX70CFvh9pO8V5yR7uGgP1x3Gpim45lLazDrdEtjDJap2TAxvTdH8b4Z_rx81Dxd_EE3AcAlpUvRfwuY" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="679" data-original-width="850" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjodNq5_5wl409M8M-p6JSIPHPG95HV3q11F1Y85rCitk7yW_4c5jxfp3VNX3nnxH6sDyYlgUVrqIA_wDRDsjsx1taBr2BjXx9JZMxdkKHZIZCGxX70CFvh9pO8V5yR7uGgP1x3Gpim45lLazDrdEtjDJap2TAxvTdH8b4Z_rx81Dxd_EE3AcAlpUvRfwuY=w400-h320" width="400" /></a></div><br /><br /></div><div><br /></div><div>laksjdlaskjd</div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-32988798014183460022023-09-03T13:44:00.005-07:002023-09-03T16:42:35.752-07:00More work towards 'visualizing' complex neural networks<div style="text-align: justify;">I'm moving quicky into a realm where the only thing that really matters are histograms showing the ditribution of data for a particular sample. And when it comes down to it, that's all a NN has, too! At this point I think that's the direction that will lead to some form of 'visualization'. I put that in quotes because it may or may not include information perceived through our eyes. It may turn out, for instance, that hearing the network is the best way to understand it, or maybe some way of understanding it through color. I have no idea, and that's why I'm doing it: because it's an interesting challenge that currently has everyone stumped. Finding something that works could lead to who knows what?</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">If I don't find anything useful, I'm still forced to completely understand NNs in order to try to represent them in form that humans can have a chance at understanding. This will make me better at designing and using NNs, which will help with my understanding, and round it goes.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Here's the sort of stuff I'm looking at. These are histogram plots of the differences between the initial weights and the trained weights for one of the dense layers (with 128 connections) of the CNN I'm using for my object ID project.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiFPy6dBJG6F4uA2uXfySesvca2j7DKzZ0xviiAKi1TndSeSVeN-z8ocDfJCblsRvEokKXOxvjmvJQJWNfvcOpxcJLZRP2XN4YdTcTzdsTmytN7mB2NSgGMl-E_SCh37bXB1edxg6rHLmMPxHBlX5uKr-5wKVgqsJQXQ_eOpVmlxq_DRVtQaq5IqbkagKgc" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="600" data-original-width="1200" height="200" src="https://blogger.googleusercontent.com/img/a/AVvXsEiFPy6dBJG6F4uA2uXfySesvca2j7DKzZ0xviiAKi1TndSeSVeN-z8ocDfJCblsRvEokKXOxvjmvJQJWNfvcOpxcJLZRP2XN4YdTcTzdsTmytN7mB2NSgGMl-E_SCh37bXB1edxg6rHLmMPxHBlX5uKr-5wKVgqsJQXQ_eOpVmlxq_DRVtQaq5IqbkagKgc=w400-h200" width="400" /></a></div><br /><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The x-range is -1 to 1, and the y range is 0 to 2000. I see no pattern. The fact remains, however, that what I'm seeing here is the difference between a model that will just randomly choose an ID each time (and therefore choose incorrectly most of the time), and a model that will choose the correct ID every single time.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><div>The fact that I don’t see anything (yet) is either an indication that I’m looking at it incorrectly (very likely), or the correct data isn’t being collected/measured/etc (also very likely).</div><div><br /></div><div>I'll keep trying.</div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-59661911907586492142023-08-15T17:41:00.004-07:002023-08-15T19:45:51.554-07:00New Accelerometer Data, and more NN 'mind'I've finally rigidly connected my accelerometer to my e-bike, and recording the g-force in x, y, z, and total. The signal to noise has improved greater than a factor of two. Here's a recording before the new connection, and then after the new connection. You can clearly see that the signal is stronger on the bottom plot<div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgxTcRJ7AnhCyIErqV0Wnrl_II_hMSieL-DDypCuXpyI91q6phIeXnUdqGyF-XhNYWwwjVkFjFGBsb-MSgwwZjaC3tbz79QL-MCUIpBcQNEf8g8rl-0TuPxjwkDNfLWy8Z6DX1Rgu51wSHCF0dHs-KXDS8b759B4C6XT0x0uUKU3cg5G4RTmJDZpIWy4gUo" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="543" data-original-width="1400" height="155" src="https://blogger.googleusercontent.com/img/a/AVvXsEgxTcRJ7AnhCyIErqV0Wnrl_II_hMSieL-DDypCuXpyI91q6phIeXnUdqGyF-XhNYWwwjVkFjFGBsb-MSgwwZjaC3tbz79QL-MCUIpBcQNEf8g8rl-0TuPxjwkDNfLWy8Z6DX1Rgu51wSHCF0dHs-KXDS8b759B4C6XT0x0uUKU3cg5G4RTmJDZpIWy4gUo=w400-h155" width="400" /></a></div><br />This produces spectrograms that are much more complex. Here's a spectrogram of the x data before the new connection:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgqaoLfYKLvXYqP2XgcQMFLAsY9I-FejmvvVSRQXBoJjdnuj60qYfIM_tzYxb5tEHMjwTX3-eLClQO9OQ45Sw_q7typRREhJX-beKLW_hXoGKqaysFRasuIBTPY-1WQrj2Y687fSL8YtMhDB0bTsNAVcFWDpsYSmHpqOfT4H5ootF18EpWyJW9L3VBS9iio" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="800" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEgqaoLfYKLvXYqP2XgcQMFLAsY9I-FejmvvVSRQXBoJjdnuj60qYfIM_tzYxb5tEHMjwTX3-eLClQO9OQ45Sw_q7typRREhJX-beKLW_hXoGKqaysFRasuIBTPY-1WQrj2Y687fSL8YtMhDB0bTsNAVcFWDpsYSmHpqOfT4H5ootF18EpWyJW9L3VBS9iio=w257-h400" width="257" /></a></div><br /><br /></div><div>... and then after the new connection:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgIIZ1Oh6e34Irh-FZUWABaw800SMAm3X49i-nXpHIu-CqoSpHjO14kxHJCpCvahg2jWbWB9phdDeUS39LMPLkgCc-QHNCGShAPYJQZI68cx912cfbEe0ozrdhA8f-ZIk_F12di5NjxQtwixCsODb205L-KCQ39ts5iQ-rLV56F2QglYDA3H9WXhByB1Ug2" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="800" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEgIIZ1Oh6e34Irh-FZUWABaw800SMAm3X49i-nXpHIu-CqoSpHjO14kxHJCpCvahg2jWbWB9phdDeUS39LMPLkgCc-QHNCGShAPYJQZI68cx912cfbEe0ozrdhA8f-ZIk_F12di5NjxQtwixCsODb205L-KCQ39ts5iQ-rLV56F2QglYDA3H9WXhByB1Ug2=w257-h400" width="257" /></a></div><br />I have no idea what all of those peaks are, and why they wander around so much. I can still see the 6 Hz line (which is the tire rotation), which in this case looks pretty steady -- so those wiggly lines probably aren't correlated with speed. Some lines go up while at the same time others go down, so I have no idea what I'm seeing. Obviously vibrations of some sort, but I'm not sure where they're coming from, and most likely they're coming from multiple sources.</div><div><br /></div><div>Here's the same two data sets, but looking at the y-axis data (this one surprised me a little):</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg255LrmWmtvdUW55cxiD6GhsEFjFcjExAnUFnXO9gznE8pxiu0Pb2Tgrn9fAh_lO28bMqJAcOn9vRUwAMolvZuBW9P0Mq7sj8RML0yqJnkKujOrWbcL3KoQSmKW5g1j4HXhL9cKd504BClI299wvME9e5jEHDp93dgYVIfSkwb5aV1RXs51vWyECCcvRDn" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="800" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEg255LrmWmtvdUW55cxiD6GhsEFjFcjExAnUFnXO9gznE8pxiu0Pb2Tgrn9fAh_lO28bMqJAcOn9vRUwAMolvZuBW9P0Mq7sj8RML0yqJnkKujOrWbcL3KoQSmKW5g1j4HXhL9cKd504BClI299wvME9e5jEHDp93dgYVIfSkwb5aV1RXs51vWyECCcvRDn=w257-h400" width="257" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgQatPnWASPSGYt3Nmb_vo2HheOTOca-xNNknMxofZoa_3qKXB9J7aCg-H6VTTOXqJT5b96T_JP03FCM6sN8fl5nsceu9hBtG1b1GHGPN-D_HMKzlAte_N_nEZUcZfR1rA5kaMcyyWFtuyYGtb_RquQk6yfzfB5mgPqp-RHz7qo2yqc7INvdGmAEurKPb4b" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="800" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEgQatPnWASPSGYt3Nmb_vo2HheOTOca-xNNknMxofZoa_3qKXB9J7aCg-H6VTTOXqJT5b96T_JP03FCM6sN8fl5nsceu9hBtG1b1GHGPN-D_HMKzlAte_N_nEZUcZfR1rA5kaMcyyWFtuyYGtb_RquQk6yfzfB5mgPqp-RHz7qo2yqc7INvdGmAEurKPb4b=w257-h400" width="257" /></a></div><br /><br />I'll be collecting the acclerometer data with this configuration for the forseeable future. The S/N is great and the spectra are baffling!</div><div><br /></div><div>In other news, I'm starting to convince myself that grabbing 65536 points from every 128 values is the correct way to go since that the structure of the python arrays and the output model (via layer.save_model()). There are still strange patterns, but some of those are probably residuals from the convolutions and pooling in previous layers. Anyhow, maybe this is pretty convincing:</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEh0kNkWY2xn2uaCrBXoFJnCJYkTqCuEK1V5-b2CrH-Sd485D9IqxIga_yqBC3eda7umNci3Ep4MOP7kCt0VwQHuovKjyW66pwQ-YjWwEH2QqjYdaERL2LYGDRb5TMrTURI-o5Z9JR8LO9-2llHqugqmKarXVD_HgH3JiLEQP3FhbtZTpQaj2ZY7G7V7lKMZ" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="512" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEh0kNkWY2xn2uaCrBXoFJnCJYkTqCuEK1V5-b2CrH-Sd485D9IqxIga_yqBC3eda7umNci3Ep4MOP7kCt0VwQHuovKjyW66pwQ-YjWwEH2QqjYdaERL2LYGDRb5TMrTURI-o5Z9JR8LO9-2llHqugqmKarXVD_HgH3JiLEQP3FhbtZTpQaj2ZY7G7V7lKMZ=w400-h400" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiJGZSLhxL4lhywWnntEJw6zCtIcM2lKXcqsmYEBml4eAZDWRv1QbO747xqDyBLa72b1CMh_glzlACFYqJRnE2FWroxJl1JPAt5Ws5mc9b0B1FqlhDx6c_ekbbg3nd0WJyNFyUJ3XU3o5h5NpA3HzIpaXn2_c6biLx_iLzPhBVjjDsCabsr5Bzg3apLDMhT" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="512" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEiJGZSLhxL4lhywWnntEJw6zCtIcM2lKXcqsmYEBml4eAZDWRv1QbO747xqDyBLa72b1CMh_glzlACFYqJRnE2FWroxJl1JPAt5Ws5mc9b0B1FqlhDx6c_ekbbg3nd0WJyNFyUJ3XU3o5h5NpA3HzIpaXn2_c6biLx_iLzPhBVjjDsCabsr5Bzg3apLDMhT=w400-h400" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEihtVOHtOyRdAU770SpU5861j0ciBX2-lfn0OXhAT_dIgsx-3OfenZhrx29Cfk9Njpxtctzl2KH238KK4chtPOHOpe6_DuMXO_E45QTcgHM7VbXMZmF5iJKXj1Y2z-B9mCUcKM8ocFHRAKPTgPMiS31MHbKHrYunLKMpJYp0JoQ7og2oqF_UUh02gvv_pPj" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="512" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEihtVOHtOyRdAU770SpU5861j0ciBX2-lfn0OXhAT_dIgsx-3OfenZhrx29Cfk9Njpxtctzl2KH238KK4chtPOHOpe6_DuMXO_E45QTcgHM7VbXMZmF5iJKXj1Y2z-B9mCUcKM8ocFHRAKPTgPMiS31MHbKHrYunLKMpJYp0JoQ7og2oqF_UUh02gvv_pPj=w400-h400" width="400" /></a></div><br /><br /><div>so, maybe this is correct. Gotta convince myself somehow. I'll look at the weights from the other models I have and see what they can show me. For now, these are weights created from synthetic image training (ahem, I think).</div></div><div><br /></div><div>Here's what I said on my discord server:</div><div><br /></div><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><div style="text-align: left;">"An attempted visualization of one set of weights of one 'dense' layer of my fully trained multi-layer convolutional neural network. The white borders are not part of the model: they represent the structure of the input data to this layer (in this case, 64 32x32 'images'). The weights shown here are applied to the input data (via a simple dot product), which results in a single data value as an output. This output (along with all other outputs in this layer) is then used as input for the next layer in the network."</div></blockquote><div><br /></div><div><br /></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-74152869411376446802023-08-14T12:22:00.000-07:002023-08-14T12:22:02.001-07:00Looking at the 'mind' of a Neural Network<div style="text-align: left;">This is an ongoing project, but since I've got a nicely trained convolutional neural network sitting around, I'm trying to figure out what's going on under the hood. No one seems to know, so I figure that's a pretty good spot for me.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">First trick is to gain access to the weights stored in a trained CNN. Once I have these, I can 'display' them in various forms to see if I can pick out any discernible patterns that might lead me to a breakthrough. The format of the input weight file took a little while to figure out, and I'm pretty sure I'm reading it in correctly. The output 'image' might be where I'm currently going wrong.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">What bothers me about what I'm currently seeing is that there are patterns that probably shouldn't be there. Here's a good example:</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEh6vd9qN3T4zHZQwXCBeARlKKQILf1HENBwvJz43EoGxhQwWb7UzDd_5hBStps0iH5qC-cVvLu6p31pcqfw5tX7lHGJ3_X9vZR1qxNloAmEq4c-VuaioTklq0FH_0iuC4nhaIHj8lr5ygaDaIEVIxqnLz5jwyhUsMcRTPms85eVcRPeT1nI8FtRH7z9swsX" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="512" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEh6vd9qN3T4zHZQwXCBeARlKKQILf1HENBwvJz43EoGxhQwWb7UzDd_5hBStps0iH5qC-cVvLu6p31pcqfw5tX7lHGJ3_X9vZR1qxNloAmEq4c-VuaioTklq0FH_0iuC4nhaIHj8lr5ygaDaIEVIxqnLz5jwyhUsMcRTPms85eVcRPeT1nI8FtRH7z9swsX=w400-h400" width="400" /></a></div><br /><br /></div><div style="text-align: left;">I'm pretty sure the similarities in columns is indicative of a read/write problem. I'll figure it out, but ugh.</div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-43515552424900075252023-08-13T18:01:00.008-07:002023-08-13T18:40:02.599-07:00Audio recording of e-bike ride<div style="text-align: left;">Here's a spectrogram of some of the audio of my ride on my e-bike today:</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjZNFbGbKZlMWdVHUp4t5mGkRWdKjhI7fAd82wXItKqmLuTUmob97pwVhLWvcmUn7TyxpHNt42UzRaMzq42vRRdlgiNS3bWn7R-fm4sBn0ZaJi3eCdKk-6Mw_M7ur0VWjam76vmpnnwYlcp-ugXdCyXtt9ta5PBbTLiw7R4tMVByP57khDCMnh_uP7WBq0_" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="675" data-original-width="847" height="319" src="https://blogger.googleusercontent.com/img/a/AVvXsEjZNFbGbKZlMWdVHUp4t5mGkRWdKjhI7fAd82wXItKqmLuTUmob97pwVhLWvcmUn7TyxpHNt42UzRaMzq42vRRdlgiNS3bWn7R-fm4sBn0ZaJi3eCdKk-6Mw_M7ur0VWjam76vmpnnwYlcp-ugXdCyXtt9ta5PBbTLiw7R4tMVByP57khDCMnh_uP7WBq0_=w400-h319" width="400" /></a></div><br /><br /></div><div style="text-align: left;">That wiggly line bothers me. I can hear it in the recording, and it's definitely the electric motor making that sound. The other thing is that I think either I've got the gain on the recorder too low (it's set to '20' for this data set), or having the recorder in my backpack is muffling (attenuating) the sound so much that the signal isn't very strong. When I listen to the recording, it sounds like it's peaking, although the data doesn't indicate that.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Here's about the first four minutes of my outbound journey:</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhMjwVQP7wUkIgg46Fe4K-p5k8fi_EYO21gyLZQWvh_M1j4GhNnSXnDZ2ou7N6puaOMLfwjtRhKoLRKFamzm7EqUhcLglfrK4IyDnG9dKqzx9QhG0mjkCV1KkbFZsBULUmsE6BjBsWsT4X2d2HoiA7RRkL1Nw64frzLri7NdNtNH4Tjokr7_RRkk9FuzFnN" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="673" data-original-width="847" height="318" src="https://blogger.googleusercontent.com/img/a/AVvXsEhMjwVQP7wUkIgg46Fe4K-p5k8fi_EYO21gyLZQWvh_M1j4GhNnSXnDZ2ou7N6puaOMLfwjtRhKoLRKFamzm7EqUhcLglfrK4IyDnG9dKqzx9QhG0mjkCV1KkbFZsBULUmsE6BjBsWsT4X2d2HoiA7RRkL1Nw64frzLri7NdNtNH4Tjokr7_RRkk9FuzFnN=w400-h318" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Also some accelerometer data with the recording device connected to the bike (as opposed to in my backback as it has been). This is a short trip -- about 2.5 miles out, and 2.5 miles in. You can clearly see the three times I hit the brakes near the beginning and end, and in the middle when I turned around. Round trip is 789.5 seconds.</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiFTWzdVl3hA5OL8aqHTahJ76-TpbvpT5S-lV8jXWJ9JbdFBPGSQJ8_We00wc6Kj1LRM3VoIRtioDV7equaFX8t4adNil0kvAi5uxAwOQS_b6MSSoseI-ZDt43BpesBnDtcGQd076pEjvThGxXLvFAiC2s39Zsr5_ncBP77YgKJg5yDxMVpMxZchaqYSojz" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="800" data-original-width="655" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEiFTWzdVl3hA5OL8aqHTahJ76-TpbvpT5S-lV8jXWJ9JbdFBPGSQJ8_We00wc6Kj1LRM3VoIRtioDV7equaFX8t4adNil0kvAi5uxAwOQS_b6MSSoseI-ZDt43BpesBnDtcGQd076pEjvThGxXLvFAiC2s39Zsr5_ncBP77YgKJg5yDxMVpMxZchaqYSojz=w328-h400" width="328" /></a></div><br />is this any better than what I already had? Not sure yet. I'm thinking that the device attached to the bike would have better S/N than the device flopping around in a backback. But this is definitely different than what i've seen before. The 6Hz line is still there, but there are other things at lower frequencies, and some interesting high freq stuff....</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">Also, Perseids:</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEh3X3wZ2ooUqL81OZnv81W_aMsfr9_9DHJlcaSImhrOV8QQmpSXbbKNC_Kqpu3rmlS9igNZW3F7f7LtEAJqfoBlJs4wHJ0K5fkGPrji5J5gzEUcPesddbCL-VzvipCmjtIKkNoDcRQACLitDbChI02Dd0uaRAM1UWKfktPj5EjY_ombA2-OPWuFNoYA6AK8" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="720" data-original-width="1280" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEh3X3wZ2ooUqL81OZnv81W_aMsfr9_9DHJlcaSImhrOV8QQmpSXbbKNC_Kqpu3rmlS9igNZW3F7f7LtEAJqfoBlJs4wHJ0K5fkGPrji5J5gzEUcPesddbCL-VzvipCmjtIKkNoDcRQACLitDbChI02Dd0uaRAM1UWKfktPj5EjY_ombA2-OPWuFNoYA6AK8=w400-h225" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhohf30vzlnsGY98g4uQbcr080GX027JxvrZS8_XWWGMGCMqtxgzolx64W9aVknRwauo1DV1suYzz3tYma9tX19-YUUQ7yuAWT6QRr3CyFNPukesvxKRGv8tOrUpvWlIjgvsMGeldXd8IRVJ8_Mj2kpct9Jv-kE5zbW53JGjXDl4B7T3p3MWNgmUWGPUW5G" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="720" data-original-width="1280" height="225" src="https://blogger.googleusercontent.com/img/a/AVvXsEhohf30vzlnsGY98g4uQbcr080GX027JxvrZS8_XWWGMGCMqtxgzolx64W9aVknRwauo1DV1suYzz3tYma9tX19-YUUQ7yuAWT6QRr3CyFNPukesvxKRGv8tOrUpvWlIjgvsMGeldXd8IRVJ8_Mj2kpct9Jv-kE5zbW53JGjXDl4B7T3p3MWNgmUWGPUW5G=w400-h225" width="400" /></a></div><br /><br /></div><div class="separator" style="clear: both; text-align: justify;">USL012, and USL011.</div></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0tag:blogger.com,1999:blog-5279763544839610047.post-81865805122417513052023-08-10T11:56:00.001-07:002023-08-10T17:26:29.654-07:00Random Thoughts #1<div style="text-align: left;"> <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEitFDaCCW1zP8S8WYLXUVtyVvzxRtdfC9KLGr2EMZiVLXml2uBeDlGV9aMCRa6e179rp04jYX5dLq6gNsCoQuERfhs1NpVnqC8sYULVL45a3FU_gvnVple7quzf0-ENMFKKyKpNeZp2Z2y_-Nj9TFYNfWvbIfWK7paQaLpMaIQ-qtmX19UWICwvuPZ0v6g0" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1871" data-original-width="2646" height="283" src="https://blogger.googleusercontent.com/img/a/AVvXsEitFDaCCW1zP8S8WYLXUVtyVvzxRtdfC9KLGr2EMZiVLXml2uBeDlGV9aMCRa6e179rp04jYX5dLq6gNsCoQuERfhs1NpVnqC8sYULVL45a3FU_gvnVple7quzf0-ENMFKKyKpNeZp2Z2y_-Nj9TFYNfWvbIfWK7paQaLpMaIQ-qtmX19UWICwvuPZ0v6g0=w400-h283" width="400" /></a></div><br /></div><div style="text-align: left;">cool graphic -- a nice reminder about what exactly pH (and other pX) is.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">... and in other news, I am now correctly (as far as I can tell) 'viewing' the weights of a trained convolutional neural network model:</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhlCTSCNR2moMJOeBYKLvqQikxu2RRy_gOnqOp_82s1j4iZNH6dO65dN6MSGjGAOVbA-KkfQkZ8QDVoZRmqBynMuCsvJpwXXR2TkA3frtNCT-YOq4q7CwqvmyOc-RR2VHBDffifYiTA1kiGQdmYfKOmdXn5FcOPgM4Y4t3d1cek7jh-ufGTdDxR3iG_ORCA" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="512" data-original-width="512" height="400" src="https://blogger.googleusercontent.com/img/a/AVvXsEhlCTSCNR2moMJOeBYKLvqQikxu2RRy_gOnqOp_82s1j4iZNH6dO65dN6MSGjGAOVbA-KkfQkZ8QDVoZRmqBynMuCsvJpwXXR2TkA3frtNCT-YOq4q7CwqvmyOc-RR2VHBDffifYiTA1kiGQdmYfKOmdXn5FcOPgM4Y4t3d1cek7jh-ufGTdDxR3iG_ORCA=w400-h400" width="400" /></a></div><br /><br /></div>Cosmic Lettucehttp://www.blogger.com/profile/08390993782675755349noreply@blogger.com0