
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 $U$, defined on vectors in 3-dimensional space, with the following properties:

(i) $U(a \mathbf{u} + b \mathbf{v}) = a U(\mathbf{u}) + b U(\mathbf{v})$ for all vectors $\mathbf{u}$ and $\mathbf{v},$ and for all scalars $a$ and $b.$
(ii) $U(\mathbf{u} \times \mathbf{v}) = U(\mathbf{u}) \times U(\mathbf{v})$ for all vectors $\mathbf{u}$ and $\mathbf{v}.$
(iii) $U \begin{pmatrix} 5 \\ 7 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \\ 7 \end{pmatrix}.$
(iv) $U \begin{pmatrix} -5 \\ 2 \\ 7 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \\ -2 \end{pmatrix}.$

Find $U \begin{pmatrix} 2 \\ 11 \\ 16 \end{pmatrix}.$

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