
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
Suppose a transformation $T$ satisfies the following properties:
(i) $T(a \mathbf{v} + b \mathbf{w}) = a T(\mathbf{v}) + b T(\mathbf{w})$ for all vectors $\mathbf{v}$ and $\mathbf{w},$ and for all scalars $a$ and $b.$
(ii) $T(\mathbf{v} \times \mathbf{w}) = T(\mathbf{v}) \times T(\mathbf{w})$ for all vectors $\mathbf{v}$ and $\mathbf{w}.$
(iii) $T \begin{pmatrix} 4 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}.$
(iv) $T \begin{pmatrix} -4 \\ 2 \\ 4 \end{pmatrix} = \begin{pmatrix} 3 \\ 5 \\ -2 \end{pmatrix}.$

Find $T \begin{pmatrix} 2 \\ 6 \\ 8 \end{pmatrix}.$

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