
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
\[(\mathbf{A}^7 + \mathbf{A}^5 + \mathbf{A}^3 + \mathbf{A} + \mathbf{I}) \mathbf{v} = \begin{pmatrix} 5 \\ 0 \end{pmatrix}.\]

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