
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}
user