
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 non-zero, distinct complex numbers \( r, s, t \) forming the vertices of an equilateral triangle in the complex plane such that for every complex number \( z \), 
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of \( c \).

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