
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
The transformation $U,$ taking vectors to vectors, has the following properties:

(i) \( U(a \mathbf{v} + b \mathbf{w}) = a U(\mathbf{v}) + b U(\mathbf{w}) \) for all vectors \( \mathbf{v} \) and \( \mathbf{w}, \) and for all scalars \( a \) and \( b. \) 
(ii) \( U(\mathbf{v} \times \mathbf{w}) = U(\mathbf{v}) \times U(\mathbf{w}) \) for all vectors \( \mathbf{v} \) and \( \mathbf{w}. \) 
(iii) \( U \left( \begin{pmatrix} 2 \\ 4 \\ 3 \end{pmatrix} \right) = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}. \)
(iv) \( U \left( \begin{pmatrix} -3 \\ 2 \\ 4 \end{pmatrix} \right) = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}. \)

Find \( U \left( \begin{pmatrix} 1 \\ 6 \\ 7 \end{pmatrix} \right). \)

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