
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
The matrix 
\[\mathbf{M} = \begin{pmatrix} x & 0 & z \\ y & 0 & -z \\ x & 0 & z \end{pmatrix}\] satisfies $\mathbf{M}^T \mathbf{M} = \mathbf{I}.$ Find $x^2 + y^2 + z^2.$

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