
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 $c$ be a complex number. Suppose there exist distinct complex numbers $p$, $q$, and $u$ such that for every complex number $z$, we have
\[
(z - p)(z - q)(z^2 - u) = (z - cp)(z - cq)(z^2 - cu).
\]
Compute the number of distinct possible values of $c$.

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