
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
Consider the matrix $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ modified by adding scalar $k$ times the identity matrix to it. Given that $(\mathbf{A} + k\mathbf{I})^T = (\mathbf{A} + k\mathbf{I})^{-1}$, with $\mathbf{I}$ being the identity matrix, find $a^2 + b^2 + c^2 + d^2$ assuming $k = 1$ and $\mathbf{A}$ is orthogonal.

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