solution: Let $a = x - 3$ and $b = x - 7.$  Then we can write the given equation as
  \[a^3 + b^3 = (a + b)^3.\]Expanding, we get $a^3 + b^3 = a^3 + 3a^2 b + 3ab^2 + b^3,$ so $3a^2 b + 3ab^2 = 0,$ which factors as
  \[3ab(a + b) = 0.\]Thus, $a = 0,$ $b = 0,$ or $a + b = 0.$  Then $x - 3 = 0,$ $x - 7 = 0,$ or $2x - 10 = 0.$  This gives us the roots $\boxed{3, 5, 7}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Find the roots of $(x - 3)^3 + (x -7)^3 = (2x - 10)^3.$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 3, 5, 7
extra_info.question: Find the roots of $(x - 3)^3 + (x -7)^3 = (2x - 10)^3.$
extra_info.split: test