
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
Determine $a, b, c$ such that the matrix
\[\mathbf{N} = \begin{pmatrix} a & b & c \\ b & -a & 0 \\ c & 0 & -a \end{pmatrix}\]
satisfies $\mathbf{N}^T \mathbf{N} = \mathbf{I}.$ Then find $a^2 + b^2 + c^2.$

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