
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $\mathbf{w} = \begin{pmatrix} 3 \\ -2 \\ 3 \end{pmatrix}.$  The set of vectors $\mathbf{v}$ such that
\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 6 \\ -4 \\ 6 \end{pmatrix}
\]
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 $\gcd(|A|,|B|,|C|,|D|) = 1.$

Present the answer in LaTex format: \boxed{Your answer}
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