
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
Consider a transformation $S$, defined as:
(i) $S(a \mathbf{u} + b \mathbf{v}) = a S(\mathbf{u}) + b S(\mathbf{v})$ for all vectors $\mathbf{u}$ and $\mathbf{v},$ and for all scalars $a$ and $b.$
(ii) $S(\mathbf{u} \times \mathbf{v}) = S(\mathbf{u}) \times S(\mathbf{v})$ for all vectors $\mathbf{u}$ and $\mathbf{v}.$
(iii) $S \begin{pmatrix} 9 \\ 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \\ 9 \end{pmatrix}.$
(iv) $S \begin{pmatrix} -3 \\ 9 \\ 4 \end{pmatrix} = \begin{pmatrix} 6 \\ 9 \\ -2 \end{pmatrix}.$

Determine $S \begin{pmatrix} 6 \\ 12 \\ 8 \end{pmatrix}.$

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