{
  "log_file": [
    "step_113-b_0.log",
    "step_113-b_1.log",
    "step_113-b_2.log",
    "step_113-b_3.log",
    "step_113-b_4.log",
    "step_113-b_5.log"
  ],
  "query": [
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nConvert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nConvert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$\n\n### Response:\nSEARCH[\"rectangle\", \"polar\", \"formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/counting_and_probability/5031.json\n\n#### Similar Question\nThe nine horizontal and nine vertical lines on an $8\\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$\n\n#### User Answer\nTo determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard, or ${9\\choose 2} = 36$. Similarily, there are ${9\\choose 2}$ ways to pick the vertical sides, giving us $r = 1296$ rectangles.\nFor $s$, there are $8^2$ unit squares, $7^2$ of the $2\\times2$ squares, and so on until $1^2$ of the $8\\times 8$ squares. Using the sum of squares formula, that gives us $s=1^2+2^2+\\cdots+8^2=\\dfrac{(8)(8+1)(2\\cdot8+1)}{6}=12*17=204$.\nThus $\\frac sr = \\dfrac{204}{1296}=\\dfrac{17}{108}$, and $m+n=125$.\n\n--- RESULTS END ---\n\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nConvert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nConvert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$\n\n### Response:\nSEARCH[\"rectangle\", \"polar\", \"formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/6145.json\n\n#### Similar Question\nTriangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\\overline{AB}$, vertex $Q$ on $\\overline{AC}$, and vertices $R$ and $S$ on $\\overline{BC}$. In terms of the side length $PQ = \\omega$, the area of $PQRS$ can be expressed as the quadratic polynomial\\[Area(PQRS) = \\alpha \\omega - \\beta \\omega^2.\\]\nThen the coefficient $\\beta = \\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n\n#### User Answer\nIf $\\omega = 25$, the area of rectangle $PQRS$ is $0$, so\n\\[\\alpha\\omega - \\beta\\omega^2 = 25\\alpha - 625\\beta = 0\\]\nand $\\alpha = 25\\beta$. If $\\omega = \\frac{25}{2}$, we can reflect $APQ$ over $PQ$, $PBS$ over $PS$, and $QCR$ over $QR$ to completely cover rectangle $PQRS$, so the area of $PQRS$ is half the area of the triangle. Using Heron's formula, since $s = \\frac{12 + 17 + 25}{2} = 27$,\n\\[[ABC] = \\sqrt{27 \\cdot 15 \\cdot 10 \\cdot 2} = 90\\]\nso\n\\[45 = \\alpha\\omega - \\beta\\omega^2 = \\frac{625}{2} \\beta - \\beta\\frac{625}{4} = \\beta\\frac{625}{4}\\]\nand\n\\[\\beta = \\frac{180}{625} = \\frac{36}{125}\\]\nso the answer is $m + n = 36 + 125 = 161$.\n\n--- RESULTS END ---\n\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nConvert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nConvert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$\n\n### Response:\nSEARCH[\"rectangle\", \"polar\", \"formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/precalculus/124.json\n\n#### Similar Question\nConvert the point $(1,-\\sqrt{3})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$\n\n#### User Answer\nWe have that $r = \\sqrt{1^2 + (-\\sqrt{3})^2} = 2.$  Also, if we draw the line connecting the origin and $(1,-\\sqrt{3}),$ this line makes an angle of $\\frac{5 \\pi}{3}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-2.5,0)--(2.5,0));\ndraw((0,-2.5)--(0,2.5));\ndraw(arc((0,0),2,0,300),red,Arrow(6));\ndraw((0,0)--(1,-sqrt(3)));\n\ndot((1,-sqrt(3)), red);\nlabel(\"$(1,-\\sqrt{3})$\", (1,-sqrt(3)), NE);\ndot((2,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\left( 2, \\frac{5 \\pi}{3} \\right).$\n\n--- RESULTS END ---\n\n"
  ],
  "response": [
    " SEARCH[\"rectangle\", \"polar\", \"formula\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe have that $r = \\sqrt{2^2 + (-\\sqrt{2})^2} = 2.$  Also, if we draw the line connecting the origin and $(\\sqrt{2},-\\sqrt{2}),$ this line makes an angle of $\\frac{\\pi}{4}$ with the positive $x$-axis.  Therefore, $\\theta = \\frac{\\pi}{4},$ so the polar coordinates are $\\boxed{\\left( 2, \\frac{\\pi}{4} \\right)}.$</s>",
    " SEARCH[\"rectangle\", \"polar\", \"formula\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe have that $r=\\sqrt{2}$ and $\\theta=-\\frac{\\pi}{2}$. Thus, the polar coordinates are $\\boxed{\\left(\\sqrt{2},-\\frac{\\pi}{2}\\right)}$.</s>",
    " SEARCH[\"rectangle\", \"polar\", \"formula\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe have that $r = \\sqrt{(\\sqrt{2})^2 + (-\\sqrt{2})^2} = 2$ and $\\theta = \\frac{1}{2} \\cdot \\frac{3 \\pi}{2} = \\frac{\\pi}{2},$ so the polar coordinates are $\\boxed{(2, \\frac{\\pi}{2})}.$</s>"
  ],
  "src_path": [
    "train/precalculus/431.json",
    "train/precalculus/431.json",
    "train/precalculus/431.json",
    "train/precalculus/431.json",
    "train/precalculus/431.json",
    "train/precalculus/431.json"
  ],
  "input": [
    "Convert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$",
    "Convert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$",
    "Convert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$",
    "Convert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$",
    "Convert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$",
    "Convert the point $(\\sqrt{2},-\\sqrt{2})$ in rectangular coordinates to polar coordinates.  Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$"
  ],
  "output": [
    "We have that $r = \\sqrt{(\\sqrt{2})^2 + (-\\sqrt{2})^2} = 2.$  Also, if we draw the line connecting the origin and $(\\sqrt{2},-\\sqrt{2}),$ this line makes an angle of $\\frac{7 \\pi}{4}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-2.5,0)--(2.5,0));\ndraw((0,-2.5)--(0,2.5));\ndraw(arc((0,0),2,0,315),red,Arrow(6));\ndraw((0,0)--(sqrt(2),-sqrt(2)));\n\ndot((sqrt(2),-sqrt(2)), red);\nlabel(\"$(\\sqrt{2},-\\sqrt{2})$\", (sqrt(2),-sqrt(2)), NE, UnFill);\ndot((2,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\boxed{\\left( 2, \\frac{7 \\pi}{4} \\right)}.$",
    "We have that $r = \\sqrt{(\\sqrt{2})^2 + (-\\sqrt{2})^2} = 2.$  Also, if we draw the line connecting the origin and $(\\sqrt{2},-\\sqrt{2}),$ this line makes an angle of $\\frac{7 \\pi}{4}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-2.5,0)--(2.5,0));\ndraw((0,-2.5)--(0,2.5));\ndraw(arc((0,0),2,0,315),red,Arrow(6));\ndraw((0,0)--(sqrt(2),-sqrt(2)));\n\ndot((sqrt(2),-sqrt(2)), red);\nlabel(\"$(\\sqrt{2},-\\sqrt{2})$\", (sqrt(2),-sqrt(2)), NE, UnFill);\ndot((2,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\boxed{\\left( 2, \\frac{7 \\pi}{4} \\right)}.$",
    "We have that $r = \\sqrt{(\\sqrt{2})^2 + (-\\sqrt{2})^2} = 2.$  Also, if we draw the line connecting the origin and $(\\sqrt{2},-\\sqrt{2}),$ this line makes an angle of $\\frac{7 \\pi}{4}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-2.5,0)--(2.5,0));\ndraw((0,-2.5)--(0,2.5));\ndraw(arc((0,0),2,0,315),red,Arrow(6));\ndraw((0,0)--(sqrt(2),-sqrt(2)));\n\ndot((sqrt(2),-sqrt(2)), red);\nlabel(\"$(\\sqrt{2},-\\sqrt{2})$\", (sqrt(2),-sqrt(2)), NE, UnFill);\ndot((2,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\boxed{\\left( 2, \\frac{7 \\pi}{4} \\right)}.$",
    "We have that $r = \\sqrt{(\\sqrt{2})^2 + (-\\sqrt{2})^2} = 2.$  Also, if we draw the line connecting the origin and $(\\sqrt{2},-\\sqrt{2}),$ this line makes an angle of $\\frac{7 \\pi}{4}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-2.5,0)--(2.5,0));\ndraw((0,-2.5)--(0,2.5));\ndraw(arc((0,0),2,0,315),red,Arrow(6));\ndraw((0,0)--(sqrt(2),-sqrt(2)));\n\ndot((sqrt(2),-sqrt(2)), red);\nlabel(\"$(\\sqrt{2},-\\sqrt{2})$\", (sqrt(2),-sqrt(2)), NE, UnFill);\ndot((2,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\boxed{\\left( 2, \\frac{7 \\pi}{4} \\right)}.$",
    "We have that $r = \\sqrt{(\\sqrt{2})^2 + (-\\sqrt{2})^2} = 2.$  Also, if we draw the line connecting the origin and $(\\sqrt{2},-\\sqrt{2}),$ this line makes an angle of $\\frac{7 \\pi}{4}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-2.5,0)--(2.5,0));\ndraw((0,-2.5)--(0,2.5));\ndraw(arc((0,0),2,0,315),red,Arrow(6));\ndraw((0,0)--(sqrt(2),-sqrt(2)));\n\ndot((sqrt(2),-sqrt(2)), red);\nlabel(\"$(\\sqrt{2},-\\sqrt{2})$\", (sqrt(2),-sqrt(2)), NE, UnFill);\ndot((2,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\boxed{\\left( 2, \\frac{7 \\pi}{4} \\right)}.$",
    "We have that $r = \\sqrt{(\\sqrt{2})^2 + (-\\sqrt{2})^2} = 2.$  Also, if we draw the line connecting the origin and $(\\sqrt{2},-\\sqrt{2}),$ this line makes an angle of $\\frac{7 \\pi}{4}$ with the positive $x$-axis.\n\n[asy]\nunitsize(0.8 cm);\n\ndraw((-2.5,0)--(2.5,0));\ndraw((0,-2.5)--(0,2.5));\ndraw(arc((0,0),2,0,315),red,Arrow(6));\ndraw((0,0)--(sqrt(2),-sqrt(2)));\n\ndot((sqrt(2),-sqrt(2)), red);\nlabel(\"$(\\sqrt{2},-\\sqrt{2})$\", (sqrt(2),-sqrt(2)), NE, UnFill);\ndot((2,0), red);\n[/asy]\n\nTherefore, the polar coordinates are $\\boxed{\\left( 2, \\frac{7 \\pi}{4} \\right)}.$"
  ],
  "rewards": [
    0.0,
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    0.0,
    0.0,
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  ]
}