{
  "log_file": [
    "step_130-b_0.log",
    "step_130-b_1.log",
    "step_130-b_2.log",
    "step_130-b_3.log",
    "step_130-b_4.log",
    "step_130-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:\nFind the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.\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:\nFind the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.\n\n### Response:\nSEARCH[\"\\\\frac{13x - x^2}{x + 1} \\\\left( x + \\\\frac{13 - x}{x + 1} \\\\right) = 42\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/1550.json\n\n#### Similar Question\nThere are constants $\\alpha$ and $\\beta$ such that $\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\\alpha+\\beta$?\n\n#### User Answer\nThe numerator $x^2 - 80x + 1551$ factors as $(x - 47)(x - 33)$, and the denominator $x^2 + 57x - 2970$ factors as $(x - 33)(x + 90)$, so \\[\\frac{x^2 - 80x + 1551}{x^2 + 57x - 2970} = \\frac{(x - 47)(x - 33)}{(x - 33)(x + 90)} = \\frac{x - 47}{x + 90}.\\]Then $\\alpha = 47$ and $\\beta = 90$, so $\\alpha + \\beta = 47 + 90 = 137$.\n\nWe can also solve the problem using Vieta's formulas, which states that the sum of the roots of the quadratic $ax^2 + bx + c = 0$ is $-b/a$.  The only way that the right-hand side $\\frac{x^2-80x+1551}{x^2+57x-2970}$ can simplify to the left-hand side $\\frac{x-\\alpha}{x+\\beta}$ is if $x^2-80x+1551$ and $x^2+57x-2970$ have a root in common.  Call this common root $\\gamma$.\n\nThen the roots of $x^2 - 80x + 1551 = 0$ are $\\alpha$ and $\\gamma$, so $\\alpha + \\gamma = 80$.  Similarly, the roots of $x^2 + 57x - 2970 = 0$ are $-\\beta$ and $\\gamma$, so $-\\beta + \\gamma = -57$.  Subtracting these equations, we get $\\alpha + \\beta = 80 - (-57) = 137$.\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:\nFind the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.\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:\nFind the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.\n\n### Response:\nSEARCH[\"\\\\frac{13x - x^2}{x + 1} \\\\left( x + \\\\frac{13 - x}{x + 1} \\\\right) = 42\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/1098.json\n\n#### Similar Question\nIn the graph of $\\frac{x^2+3x+2}{x^3+x^2-2x}$, let $a$ be the number of holes in the graph, $b$ be the number of vertical asympotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Find $a+2b+3c+4d$.\n\n#### User Answer\nWe can factor the numerator and denominator to get $$\\frac{x^2+3x+2}{x^3+x^2-2x} = \\frac{(x+1)(x+2)}{x(x-1)(x+2)}.$$In this representation we can immediately see that there is a hole at $x=-2$, and vertical asymptotes at $x=1$ and $x=0$. There are no more holes or vertical asymptotes, so $a=1$ and $b=2$. If we cancel out the common factors we have\n$$\\frac{(x+1)(x+2)}{x(x-1)(x+2)} =\\frac{x+1}{x^2-x}.$$We can now see that as $x$ becomes very large, the $x^2$ term in the denominator dominates and the graph tends towards $0$, giving us a horizontal asymptote.\nSince the graph cannot have more than one horizontal asymptote, or a horizontal asymptote and a slant asymptote, we have that $c=1$ and $d=0$. Therefore, $a+2b+3c+4d = 1+2\\cdot 2+3+0 = 8.$\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:\nFind the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.\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:\nFind the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.\n\n### Response:\nSEARCH[\"\\\\frac{13x - x^2}{x + 1} \\\\left( x + \\\\frac{13 - x}{x + 1} \\\\right) = 42\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/1120.json\n\n#### Similar Question\nCompute $\\frac{x^8+12x^4+36}{x^4+6}$ when $x=5$.\n\n#### User Answer\nNote that $\\left(x^4+6\\right)^2=x^8+12x^4+36$. So $\\frac{x^8+12x^4+36}{x^4+6}=\\frac{\\left(x^4+6\\right)^2}{x^4+6}=x^4+6$. Our answer is therefore $5^4+6=625+6=631$.\n\n--- RESULTS END ---\n\n"
  ],
  "response": [
    " SEARCH[\"\\\\frac{13x - x^2}{x + 1} \\\\left( x + \\\\frac{13 - x}{x + 1} \\\\right) = 42\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nLet $y = x + 1.$  Then\n\\[\\frac{13x - x^2}{y} \\left( y - \\frac{13 - x}{y} \\right) = 42.\\]Multiplying both sides by $y^2,$ we get\n\\[\\frac{13x - x^2}{y} \\left( y - \\frac{13 - x}{y} \\right) y^2 = 42y^2.\\]This simplifies to $13xy^2 - 2x^2y - 13y^2 + 13y^3 = 42y^2.$  This rearranges to $13y^2 (x - 1) - 2x^2y - 13y^2 + 13y^3 = 0,$ which simplifies to $2x^2y - 13y^3 = 0.$  This factors as $y (2x^2 - 13y^2) = 0,$ so $y = 0$ or $2x^2 = 13y^2.$  The latter equation simplifies to $x^2 = \\frac{13}{2} y^2,$ so $x = \\pm \\frac{13}{2} y.$\n\nWe want $x + 1 = y,$ so $y = x + 1.$  Then\n\\[x = \\pm \\frac{13}{2} (x + 1).\\]This simplifies to $x = \\pm 3,$ so the solutions are $\\boxed{3, -3}.$</s>",
    " SEARCH[\"\\\\frac{13x - x^2}{x + 1} \\\\left( x + \\\\frac{13 - x}{x + 1} \\\\right) = 42\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe can factor the numerator and denominator to get\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = \\frac{(x+1)(x+13)}{x(x+1)}.\\]We can then write the equation as\n\\[(x+1)(x+13) = 42x(x+1).\\]Expanding, we get $x^2 + 14x + 13 = 42x^2 + 42x.$\n\nRearranging, we get $x^2 - 29x + 13 = 0,$ which factors as $(x - 3)(x - 4) = 0.$  Therefore, the solutions are $\\boxed{3, 4}.$</s>",
    " SEARCH[\"\\\\frac{13x - x^2}{x + 1} \\\\left( x + \\\\frac{13 - x}{x + 1} \\\\right) = 42\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nLet $y = x + 1.$  Then\n\\[\\frac{13x - x^2}{x + 1} = \\frac{13y - y^2}{y} = 13 - \\frac{y^2}{y} = 13 - y.\\]Hence,\n\\[(13 - y) \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Then\n\\[13x - y^2 = 42.\\]We can factor this as\n\\[(y - 3)(13x - y) = 42.\\]If $y = 3,$ then $x = 6,$ so $(x,y) = (6,3)$ is a solution.  If $y = -3,$ then $x = -2,$ so $(x,y) = (-2,-3)$ is a solution.  If $y = 13,$ then $x = -1,$ so $(x,y) = (-1,13)$ is a solution.  If $y = -13,$ then $x = 10,$ so $(x,y) = (10,-13)$ is a solution.  Thus, the solutions are $\\boxed{(6,3), ({-2},{-3}), ({-1},13), \\text{and} (10,-13)}.$</s>"
  ],
  "src_path": [
    "train/intermediate_algebra/731.json",
    "train/intermediate_algebra/731.json",
    "train/intermediate_algebra/731.json",
    "train/intermediate_algebra/731.json",
    "train/intermediate_algebra/731.json",
    "train/intermediate_algebra/731.json"
  ],
  "input": [
    "Find the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.",
    "Find the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.",
    "Find the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.",
    "Find the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.",
    "Find the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas.",
    "Find the solutions to\n\\[\\frac{13x - x^2}{x + 1} \\left( x + \\frac{13 - x}{x + 1} \\right) = 42.\\]Enter all the solutions, separated by commas."
  ],
  "output": [
    "Multiplying both sides by $(x + 1)^2,$ we get\n\\[(13x - x^2)(x(x + 1) + (13 - x)) = 42(x + 1)^2.\\]This expands to $x^4 - 13x^3 + 55x^2 - 85x + 42 = 0,$ which factors as $(x - 1)(x - 6)(x^2 - 6x + 7) = 0.$  By the quadratic formula, the roots of $x^2 - 6x + 7 = 0$ are $3 \\pm \\sqrt{2}.$  Therefore, the solutions are $\\boxed{1, 6, 3 + \\sqrt{2}, 3 - \\sqrt{2}}.$",
    "Multiplying both sides by $(x + 1)^2,$ we get\n\\[(13x - x^2)(x(x + 1) + (13 - x)) = 42(x + 1)^2.\\]This expands to $x^4 - 13x^3 + 55x^2 - 85x + 42 = 0,$ which factors as $(x - 1)(x - 6)(x^2 - 6x + 7) = 0.$  By the quadratic formula, the roots of $x^2 - 6x + 7 = 0$ are $3 \\pm \\sqrt{2}.$  Therefore, the solutions are $\\boxed{1, 6, 3 + \\sqrt{2}, 3 - \\sqrt{2}}.$",
    "Multiplying both sides by $(x + 1)^2,$ we get\n\\[(13x - x^2)(x(x + 1) + (13 - x)) = 42(x + 1)^2.\\]This expands to $x^4 - 13x^3 + 55x^2 - 85x + 42 = 0,$ which factors as $(x - 1)(x - 6)(x^2 - 6x + 7) = 0.$  By the quadratic formula, the roots of $x^2 - 6x + 7 = 0$ are $3 \\pm \\sqrt{2}.$  Therefore, the solutions are $\\boxed{1, 6, 3 + \\sqrt{2}, 3 - \\sqrt{2}}.$",
    "Multiplying both sides by $(x + 1)^2,$ we get\n\\[(13x - x^2)(x(x + 1) + (13 - x)) = 42(x + 1)^2.\\]This expands to $x^4 - 13x^3 + 55x^2 - 85x + 42 = 0,$ which factors as $(x - 1)(x - 6)(x^2 - 6x + 7) = 0.$  By the quadratic formula, the roots of $x^2 - 6x + 7 = 0$ are $3 \\pm \\sqrt{2}.$  Therefore, the solutions are $\\boxed{1, 6, 3 + \\sqrt{2}, 3 - \\sqrt{2}}.$",
    "Multiplying both sides by $(x + 1)^2,$ we get\n\\[(13x - x^2)(x(x + 1) + (13 - x)) = 42(x + 1)^2.\\]This expands to $x^4 - 13x^3 + 55x^2 - 85x + 42 = 0,$ which factors as $(x - 1)(x - 6)(x^2 - 6x + 7) = 0.$  By the quadratic formula, the roots of $x^2 - 6x + 7 = 0$ are $3 \\pm \\sqrt{2}.$  Therefore, the solutions are $\\boxed{1, 6, 3 + \\sqrt{2}, 3 - \\sqrt{2}}.$",
    "Multiplying both sides by $(x + 1)^2,$ we get\n\\[(13x - x^2)(x(x + 1) + (13 - x)) = 42(x + 1)^2.\\]This expands to $x^4 - 13x^3 + 55x^2 - 85x + 42 = 0,$ which factors as $(x - 1)(x - 6)(x^2 - 6x + 7) = 0.$  By the quadratic formula, the roots of $x^2 - 6x + 7 = 0$ are $3 \\pm \\sqrt{2}.$  Therefore, the solutions are $\\boxed{1, 6, 3 + \\sqrt{2}, 3 - \\sqrt{2}}.$"
  ],
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