
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 \( T \), applying to vectors, is defined such that:
1. \( T(a \mathbf{u} + b \mathbf{v}) = a T(\mathbf{u}) + b T(\mathbf{v}) \) for all vectors \( \mathbf{u}, \mathbf{v} \) and scalars \( a, b \).
2. \( T(\mathbf{u} \times \mathbf{v}) = T(\mathbf{u}) \times T(\mathbf{v}) \) for all vectors \( \mathbf{u}, \mathbf{v} \).
3. \( T \begin{pmatrix} 3 \\ 3 \\ 7 \end{pmatrix} = \begin{pmatrix} 2 \\ -4 \\ 5 \end{pmatrix} \).
4. \( T \begin{pmatrix} -2 \\ 5 \\ 4 \end{pmatrix} = \begin{pmatrix} 6 \\ 1 \\ 0 \end{pmatrix} \).

Find \( T \begin{pmatrix} 5 \\ 15 \\ 11 \end{pmatrix} \).

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