
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 \[\mathbf{M} = \begin{pmatrix} a & b & 0 \\ 0 & a & b \\ b & 0 & a \end{pmatrix}\] be a matrix with complex entries such that $\mathbf{M}^3 = \mathbf{I}$. If $ab = 1$, find the possible values of $a^3 + b^3$.

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