
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 $\mathbf{A} = \begin{pmatrix} x & 3 \\ -4 & y \end{pmatrix}$ for some real numbers $x$ and $y,$ and let $\mathbf{B} = 3\mathbf{A}^{-1}$. If
\[\mathbf{A} + \mathbf{B} = \mathbf{I},\]
where $\mathbf{I}$ is the identity matrix, find $\det \mathbf{A}$.

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