
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 \( U \) acting on vectors is defined by:
(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) \( U \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} \).
(iv) \( U \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} = \begin{pmatrix} -1 \\ 4 \\ 3 \end{pmatrix} \).

Find \( U \begin{pmatrix} 2 \\ 5 \\ 7 \end{pmatrix} \).

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