solution: For the first line, let $t = 2x = 3y = -z.$  Then
  \[\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} t/2 \\ t/3 \\ -t \end{pmatrix} = \frac{t}{6} \begin{pmatrix} 3 \\ 2 \\ -6 \end{pmatrix}.\]Thus, the direction vector of the first line is $\begin{pmatrix} 3 \\ 2 \\ -6 \end{pmatrix}.$
  
  For the second line, let $t = 6x = -y = -4z.$  Then
  \[\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} t/6 \\ -t \\ -t/4 \end{pmatrix} = \frac{t}{12} \begin{pmatrix} 2 \\ -12 \\ -3 \end{pmatrix}.\]Thus, the direction vector of the first line is $\begin{pmatrix} 2 \\ -12 \\ -3 \end{pmatrix}.$
  
  Note that
  \[\begin{pmatrix} 3 \\ 2 \\ -6 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -12 \\ -3 \end{pmatrix} = 0.\]Hence, the angle between the lines is $\boxed{90^\circ}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  The set of points $(x,y,z)$ that satisfy
  \[2x = 3y = -z\]is a line.
  
  The set of points $(x,y,z)$ that satisfy
  \[6x = -y = -4z\]is another line.
  
  Find the angle between these lines, in degrees.
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 90^\circ
extra_info.question: The set of points $(x,y,z)$ that satisfy
  \[2x = 3y = -z\]is a line.
  
  The set of points $(x,y,z)$ that satisfy
  \[6x = -y = -4z\]is another line.
  
  Find the angle between these lines, in degrees.
extra_info.split: test