
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
If the matrix $\mathbf{B}$ has an inverse and $(\mathbf{B} - 3 \mathbf{I})(\mathbf{B} - 5 \mathbf{I}) = \mathbf{0},$ then find
\[\mathbf{B} + 10 \mathbf{B}^{-1}.\]

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