solution: The projection of $\begin{pmatrix} 2 \\ y \\ -5 \end{pmatrix}$ onto $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$ is
  \[\frac{\begin{pmatrix} 2 \\ y \\ -5 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}}{\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}} \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} = \frac{-2y - 3}{6} \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}.\]Then $-2y - 3 = 5,$ so $y = \boxed{-4}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  The projection of $\begin{pmatrix} 2 \\ y \\ -5 \end{pmatrix}$ onto $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$ is
  \[\frac{5}{6} \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}.\]Find $y.$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: -4
extra_info.question: The projection of $\begin{pmatrix} 2 \\ y \\ -5 \end{pmatrix}$ onto $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$ is
  \[\frac{5}{6} \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}.\]Find $y.$
extra_info.split: test