
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
\[\mathbf{N} = \begin{pmatrix} d & e & f \\ e & f & d \\ f & d & e \end{pmatrix}\]be a matrix with complex entries such that $\mathbf{N}^2 = \mathbf{I}$. If $def = -1$, find the possible values of $d^3 + e^3 + f^3$.

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