
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 \(r\), \(s\), and \(t\) such that for every complex number \(z\), we have
\[
  (z - r)(z - s)(z - t) = (z - 2cr)(z - 2cs)(z - 2ct).
\]
Compute the number of distinct possible values of \(c\).

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