
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
\[
\mathbf{M} = \begin{pmatrix} a & b & c \\ c & a & b \\ b & c & a \end{pmatrix}
\]
be a matrix with real entries such that $\mathbf{M}^2 = 2\mathbf{I}$. If $abc = -1$, find the possible values of $a^3 + b^3 + c^3.$

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