Let $\mathbf{w} = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}.$  The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 4 \\ -2 \\ 4 \end{pmatrix}\]lie on a plane.  Enter the equation of this plane in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $\mathbf{w} = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}.$  The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 4 \\ -2 \\ 4 \end{pmatrix}\]lie on a plane.  Enter the equation of this plane 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

user
2x - y + 2z - 18 = 0
Let $\mathbf{w} = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}.$  The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 4 \\ -2 \\ 4 \end{pmatrix}\]lie on a plane.  Enter the equation of this plane in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$
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