
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 $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} x & y \\ z & w \end{pmatrix}$ be two matrices such that $\mathbf{A}^T = \mathbf{A}^{-1}$ and $\mathbf{B}^T = \mathbf{B}^{-1}$. If $\mathbf{C} = \mathbf{A} \mathbf{B}$, find the value of $x^2 + y^2 + z^2 + w^2$.

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