
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
Let transformation \( T \) have the following properties:
(i) \( T(a \mathbf{v} + b \mathbf{w}) = a T(\mathbf{v}) + b T(\mathbf{w}) \) for all vectors \( \mathbf{v} \) and \( \mathbf{w}, \) and for all scalars \( a \) and \( b. \)
(ii) \( T(\mathbf{v} \times \mathbf{w}) = T(\mathbf{v}) \times T(\mathbf{w}) \) for all vectors \( \mathbf{v} \) and \( \mathbf{w}. \)
(iii) \( T \begin{pmatrix} 8 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 5 \\ -2 \\ 9 \end{pmatrix}. \)
(iv) \( T \begin{pmatrix} -8 \\ 2 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ 9 \\ -2 \end{pmatrix}. \)

Find \( T \begin{pmatrix} 4 \\ 10 \\ 14 \end{pmatrix}. \)

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