
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 $r$, $s$, $t$, and $u$ such that for every complex number $z$, we have
\[
(z - r)(z - s)(z - t)(z - u) = (z - cr)(z - cs)(z - ct)(z - cu).
\]
Compute the number of distinct possible values of $c$.

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