
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 \(\mathbf{A} = \begin{pmatrix} a & b & e \\ c & d & f \\ g & h & i \end{pmatrix}\), and its transpose is given by \(\mathbf{A}^T = \begin{pmatrix} a & c & g \\ b & d & h \\ e & f & i \end{pmatrix}\). Given that \(\mathbf{A}^T = \mathbf{A}^{-1}\), find the sum \(a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 + h^2 + i^2\).

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