
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]


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Suppose $A(z)$, $B(z)$, and $C(z)$ are polynomials with real coefficients, having degrees 4, 1, and 5 respectively, and constant terms $5$, $6$, and $42$ respectively. Determine the number of distinct complex numbers $z$ that satisfy $A(z) \cdot B(z) = C(z)$.

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