
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
Define \( \mathbf{A} = \begin{pmatrix} 0 & 1 \\ 4 & 0 \end{pmatrix} \). Find the vector \( \mathbf{v} \) such that
\[ (\mathbf{A}^4 + \mathbf{A}^2 + \mathbf{I}) \mathbf{v} = \begin{pmatrix} 0 \\ 5 \end{pmatrix}. \]

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