
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 $a$ and $b$ be complex numbers, and $r, s, t$ distinct complex roots such that the polynomial $p(z) = (z - r)(z - s)(z - t)$ is transformed under $z \to az + b$ to $q(z) = (z - cr)(z - cs)(z - ct)$. Find the number of distinct possible values of $c$ under this transformation.

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