
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 \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w}\) be vectors such that
\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} -1 \\ 2 \\ -2 \end{pmatrix}
\]
and 
\[
\operatorname{proj}_{\mathbf{w}} \mathbf{u} = \begin{pmatrix} 3 \\ 0 \\ -4 \end{pmatrix}.
\]
Compute \(\operatorname{proj}_{\mathbf{w}} (3 \mathbf{v} - \mathbf{u})\).

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