
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 the mutually orthogonal vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 3 \\ -4 \\ 1 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} 5 \\ 2 \\ -6 \end{pmatrix}$, find the scalars $p$, $q$, and $r$ such that:
\[
\begin{pmatrix} -5 \\ 10 \\ 4 \end{pmatrix} = p \mathbf{a} + q \mathbf{b} + r \mathbf{c}.
\]
Enter the ordered triple $(p,q,r)$.

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