
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
A transformation $S,$ operating on vectors, has the following properties:
(i) $S(a \mathbf{v} + b \mathbf{w}) = a S(\mathbf{v}) + b S(\mathbf{w})$ for all vectors $\mathbf{v}$ and $\mathbf{w},$ and for all scalars $a$ and $b.$
(ii) $S(\mathbf{v} \times \mathbf{w}) = S(\mathbf{v}) \times S(\mathbf{w})$ for all vectors $\mathbf{v}$ and $\mathbf{w}.$
(iii) $S \begin{pmatrix} 8 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 5 \\ 0 \\ 10 \end{pmatrix}.$
(iv) $S \begin{pmatrix} -8 \\ 2 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ 10 \\ 0 \end{pmatrix}.$

Determine $S \begin{pmatrix} 4 \\ 10 \\ 12 \end{pmatrix}.$

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