solution: Let $\mathbf{a} = \begin{pmatrix} 0 \\ -1 \\ -1 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} -4 \\ 4 \\ 4 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 4 \\ 5 \\ 1 \end{pmatrix}.$  Then the normal vector of the plane is orthogonal to both
  \[\mathbf{b} - \mathbf{a} = \begin{pmatrix} -4 \\ 5 \\ 5 \end{pmatrix}\]and
  \[\mathbf{c} - \mathbf{a} = \begin{pmatrix} 4 \\ 6 \\ 2 \end{pmatrix}.\]So to compute the normal vector, we take the cross product of these vectors:
  \[\begin{pmatrix} -4 \\ 5 \\ 5 \end{pmatrix} \times \begin{pmatrix} 4 \\ 6 \\ 2 \end{pmatrix} = \begin{pmatrix} -20 \\ 28 \\ -44 \end{pmatrix}.\]We can scale this vector, and take $\begin{pmatrix} 5 \\ -7 \\ 11 \end{pmatrix}$ as the normal vector.  Then the equation of the plane is of the form
  \[5x - 7y + 11z + D = 0.\]Substituting the coordinates of any of the points, we find that the equation of the plane is $\boxed{5x - 7y + 11z + 4 = 0}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Find the equation of the plane containing the points $(0,-1,-1),$ $(-4,4,4),$ and $(4,5,1).$  Enter your answer in the form
  \[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 5x - 7y + 11z + 4 = 0
extra_info.question: Find the equation of the plane containing the points $(0,-1,-1),$ $(-4,4,4),$ and $(4,5,1).$  Enter your answer in the form
  \[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$
extra_info.split: test