
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
The vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix},$ $\mathbf{d} = \begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}$ are mutually orthogonal. There exist scalars $s,$ $t,$ and $u$ such that
\[\begin{pmatrix} 8 \\ -4 \\ 0 \end{pmatrix} = s \mathbf{a} + t \mathbf{d} + u \mathbf{c}.\]Enter the ordered triple $(s,t,u)$.

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