
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
Given that the vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix},$ $\mathbf{v} = \begin{pmatrix} 3 \\ -6 \\ 1 \end{pmatrix},$ and $\mathbf{w} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$ are mutually orthogonal, find scalars $x,$ $y,$ and $z$ such that
\[
\begin{pmatrix} 5 \\ -7 \\ 4 \end{pmatrix} = x \mathbf{u} + y \mathbf{v} + z \mathbf{w}.
\]

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