
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
\[\mathbf{N} = \begin{pmatrix} p & q & r \\ q & r & p \\ r & p & q \end{pmatrix}\]
be a matrix with complex entries such that \(\mathbf{N}^3 = \mathbf{I}\). If \(pqr = -1\), find the possible values of \(p^3 + q^3 + r^3\).

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