
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 $\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -4 & 0 \end{pmatrix}$. Find the vector $\mathbf{u}$ such that
\[(\mathbf{B}^{10} + \mathbf{B}^8 + \mathbf{B}^6 + \mathbf{B}^4 + \mathbf{B}^2 + \mathbf{I}) \mathbf{u} = \begin{pmatrix} 0 \\ -20 \end{pmatrix}.\]

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