
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
Given a transformation $U$, define the following properties for vectors $\mathbf{u}$ and $\mathbf{v}$ and scalars $c$ and $d$:
(i) $U(c \mathbf{u} + d \mathbf{v}) = c U(\mathbf{u}) + d U(\mathbf{v})$.
(ii) $U(\mathbf{u} \times \mathbf{v}) = U(\mathbf{u}) \times U(\mathbf{v})$.
(iii) $U \begin{pmatrix} 4 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix}$.
(iv) $U \begin{pmatrix} -4 \\ 2 \\ 4 \end{pmatrix} = \begin{pmatrix} 3 \\ 6 \\ -1 \end{pmatrix}$.

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

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