
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
Define a plane $Q$ that passes through the origin and has a normal vector $\mathbf{n} = \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$. Find the matrix $\mathbf{S}$ such that for any vector $\mathbf{u}$, $\mathbf{S} \mathbf{u}$ is the reflection of $\mathbf{u}$ through plane $Q$.

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