solution: Since $z^5 = 1,$ $z^5 - 1 = 0,$ which factors as
  \[(z - 1)(z^4 + z^3 + z^2 + z + 1) = 0.\]Since $z \neq 1,$ $z^4 + z^3 + z^2 + z + 1 = 0.$
  
  Then
  \[z + \frac{1}{z} + z^2 + \frac{1}{z^2} = \frac{z^3 + z + z^4 + 1}{z^2} = \frac{-z^2}{z^2} = \boxed{-1}.\]
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Let $z$ be a complex number such that $z^5 = 1$ and $z \neq 1.$  Compute
  \[z + \frac{1}{z} + z^2 + \frac{1}{z^2}.\]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: -1
extra_info.question: Let $z$ be a complex number such that $z^5 = 1$ and $z \neq 1.$  Compute
  \[z + \frac{1}{z} + z^2 + \frac{1}{z^2}.\]
extra_info.split: test