
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 \(\mathbf{B} = \begin{pmatrix} b & 2 \\ -3 & e \end{pmatrix}\) for some real numbers \(b\) and \(e\). If
\[\mathbf{B} + \mathbf{B}^{-1} = \mathbf{I},\]
where \(\mathbf{I}\) is the identity matrix, then find \(\det \mathbf{B}\).

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