Update README.md

README.md

@@ -12,6 +12,92 @@ language:
 - en
 ---
 
+# Overview
+This model was trained using Phase 2 data of PRM800K. 
+It first outputs thoughts in a "thinking" message, with each thought delimited by a `<|end_of_thought|>` token. 
+Then, it makes an assistant message containing the final answer.
+
+Example:
+```
+Question: Find the largest possible real part of\[(75+117i)z+\frac{96+144i}{z}\]where $z$ is a complex number with $|z|=4$.
+
+Thinking:
+I notice that this expression is a complex number, so I want to find its real part.<|end_of_thought|>
+
+To do that, I need to simplify the expression first.<|end_of_thought|>
+
+I can use the fact that $|z|=4$ to rewrite $z$ in polar form: $z=4e^{i\theta}$ where $\theta$ is some angle.<|end_of_thought|>
+
+Then, I can use the distributive property and the fact that $i^2=-1$ to expand the expression: \[(75+117i)z+\frac{96+144i}{z}=(75+117i)(4e^{i\theta})+\frac{96+144i}{4e^{i\theta}}\] \[=300e^{i\theta}+468ie^{i\theta}+24e^{-i\theta}+36ie^{-i\theta}\]<|end_of_thought|>
+
+Now, I can use the fact that $e^{ix}=\cos x+i\sin x$ and $e^{-ix}=\cos x-i\sin x$ to write the expression in terms of trigonometric functions: \[=300\cos\theta+468i\sin\theta+24\cos\theta-36i\sin\theta\] \[=(324\cos\theta+432\sin\theta)+i(432\sin\theta-324\cos\theta)\]<|end_of_thought|>
+
+The real part of this expression is $324\cos\theta+432\sin\theta$, which is a linear combination of $\cos\theta$ and $\sin\theta$.<|end_of_thought|>
+
+I know that the maximum value of a linear combination of $\cos\theta$ and $\sin\theta$ occurs when the coefficients are proportional to the direction of the vector $(\cos\theta,\sin\theta)$.<|end_of_thought|>
+
+In other words, the maximum value occurs when the expression is equal to the length of the vector times the unit vector in the direction of the vector.<|end_of_thought|>
+
+The length of the vector $(\cos\theta,\sin\theta)$ is $\sqrt{\cos^2\theta+\sin^2\theta}=1$, so the maximum value is just the length of the coefficients: $|324\cos\theta+432\sin\theta|$.<|end_of_thought|>
+
+To find the maximum value, I need to find the angle that makes the expression equal to the length of the coefficients.<|end_of_thought|>
+
+I can use the dot product formula to find this angle: \[(324\cos\theta+432\sin\theta)\cdot (1,1)=(324,432)\cdot (\cos\theta,\sin\theta)=324\cos\theta+432\sin\theta\]<|end_of_thought|>
+
+The dot product of two vectors is equal to the product of their lengths and the cosine of the angle between them.<|end_of_thought|>
+
+So, I have \[324\cos\theta+432\sin\theta=|324,432|\cos\theta=|324,432|\cos\phi\] where $\phi$ is the angle between the vectors $(324,432)$ and $(\cos\theta,\sin\theta)$.<|end_of_thought|>
+
+I can use the inverse cosine function to solve for $\theta$: \[\theta=\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\]<|end_of_thought|>
+
+Now, I can plug this value of $\theta$ into the expression for the real part and simplify: \[324\cos\theta+432\sin\theta=324\cos\left(\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\right)+432\sin\left(\cos^{-1}\left(\frac{324\cos\phi+432\sin\phi}{|324,432|}\right)\right)\] \[=324\frac{324\cos\phi+432\sin\phi}{|324,432|}+432\frac{-324\sin\phi+432\cos\phi}{|324,432|}\] \[=\frac{324^2\cos\phi+432^2\sin\phi+432^2\cos\phi-324^2\sin\phi}{|324,432|}\] \[=\frac{(324^2+432^2)\cos\phi+(432^2-324^2)\sin\phi}{|324,432|}\]<|end_of_thought|>
+
+I can factor out the common factor of $324^2$ from the numerator and simplify the expression: \[=\frac{324^2(\cos\phi+\sin\phi)+432^2(\sin\phi-\cos\phi)}{|324,432|}\] \[=\frac{324^2\cos\phi+432^2\sin\phi}{|324,432|}\]<|end_of_thought|>
+
+This is the maximum possible real part of the expression, and it occurs when $\phi$ is the angle between the vectors $(324,432)$ and $(\cos\theta,\sin\theta)$.<|end_of_thought|>
+
+Final output:
+# Answer
+
+\frac{324^2\cos\phi+432^2\sin\phi}{|324,432|}
+```
+
+This is how the message format works (you do apply_chat_template on it to get prompt)
+```json
+[
+   {
+      "role":"user",
+      "content":"Find the largest possible real part of\\[(75+117i)z+\\frac{96+144i}{z}\\]where $z$ is a complex number with $|z|=4$."
+   },
+   {
+      "role":"thinking",
+      "content":[
+         "I notice that this expression is a complex number, so I want to find its real part.",
+         "To do that, I need to simplify the expression first.",
+         "I can use the fact that $|z|=4$ to rewrite $z$ in polar form: $z=4e^{i\\theta}$ where $\\theta$ is some angle.",
+         "Then, I can use the distributive property and the fact that $i^2=-1$ to expand the expression: \\[(75+117i)z+\\frac{96+144i}{z}=(75+117i)(4e^{i\\theta})+\\frac{96+144i}{4e^{i\\theta}}\\] \\[=300e^{i\\theta}+468ie^{i\\theta}+24e^{-i\\theta}+36ie^{-i\\theta}\\]",
+         "Now, I can use the fact that $e^{ix}=\\cos x+i\\sin x$ and $e^{-ix}=\\cos x-i\\sin x$ to write the expression in terms of trigonometric functions: \\[=300\\cos\\theta+468i\\sin\\theta+24\\cos\\theta-36i\\sin\\theta\\] \\[=(324\\cos\\theta+432\\sin\\theta)+i(432\\sin\\theta-324\\cos\\theta)\\]",
+         "The real part of this expression is $324\\cos\\theta+432\\sin\\theta$, which is a linear combination of $\\cos\\theta$ and $\\sin\\theta$.",
+         "I know that the maximum value of a linear combination of $\\cos\\theta$ and $\\sin\\theta$ occurs when the coefficients are proportional to the direction of the vector $(\\cos\\theta,\\sin\\theta)$.",
+         "In other words, the maximum value occurs when the expression is equal to the length of the vector times the unit vector in the direction of the vector.",
+         "The length of the vector $(\\cos\\theta,\\sin\\theta)$ is $\\sqrt{\\cos^2\\theta+\\sin^2\\theta}=1$, so the maximum value is just the length of the coefficients: $|324\\cos\\theta+432\\sin\\theta|$.",
+         "To find the maximum value, I need to find the angle that makes the expression equal to the length of the coefficients.",
+         "I can use the dot product formula to find this angle: \\[(324\\cos\\theta+432\\sin\\theta)\\cdot (1,1)=(324,432)\\cdot (\\cos\\theta,\\sin\\theta)=324\\cos\\theta+432\\sin\\theta\\]",
+         "The dot product of two vectors is equal to the product of their lengths and the cosine of the angle between them.",
+         "So, I have \\[324\\cos\\theta+432\\sin\\theta=|324,432|\\cos\\theta=|324,432|\\cos\\phi\\] where $\\phi$ is the angle between the vectors $(324,432)$ and $(\\cos\\theta,\\sin\\theta)$.",
+         "I can use the inverse cosine function to solve for $\\theta$: \\[\\theta=\\cos^{-1}\\left(\\frac{324\\cos\\phi+432\\sin\\phi}{|324,432|}\\right)\\]",
+         "Now, I can plug this value of $\\theta$ into the expression for the real part and simplify: \\[324\\cos\\theta+432\\sin\\theta=324\\cos\\left(\\cos^{-1}\\left(\\frac{324\\cos\\phi+432\\sin\\phi}{|324,432|}\\right)\\right)+432\\sin\\left(\\cos^{-1}\\left(\\frac{324\\cos\\phi+432\\sin\\phi}{|324,432|}\\right)\\right)\\] \\[=324\\frac{324\\cos\\phi+432\\sin\\phi}{|324,432|}+432\\frac{-324\\sin\\phi+432\\cos\\phi}{|324,432|}\\] \\[=\\frac{324^2\\cos\\phi+432^2\\sin\\phi+432^2\\cos\\phi-324^2\\sin\\phi}{|324,432|}\\] \\[=\\frac{(324^2+432^2)\\cos\\phi+(432^2-324^2)\\sin\\phi}{|324,432|}\\]",
+         "I can factor out the common factor of $324^2$ from the numerator and simplify the expression: \\[=\\frac{324^2(\\cos\\phi+\\sin\\phi)+432^2(\\sin\\phi-\\cos\\phi)}{|324,432|}\\] \\[=\\frac{324^2\\cos\\phi+432^2\\sin\\phi}{|324,432|}\\]",
+         "This is the maximum possible real part of the expression, and it occurs when $\\phi$ is the angle between the vectors $(324,432)$ and $(\\cos\\theta,\\sin\\theta)$."
+      ]
+   },
+   {
+      "role":"assistant",
+      "content":"# Answer\n\n\\frac{324^2\\cos\\phi+432^2\\sin\\phi}{|324,432|}"
+   }
+]
+```
+
 # Uploaded  model
 
 - **Developed by:** Broyojo