
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
Suppose that $P(z), Q(z), R(z),$ and $S(z)$ are polynomials with real coefficients, having degrees 4, 3, 5, and 1, respectively, and constant terms 5, 2, 10, and 1, respectively. Determine the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) + S(z) = R(z)$.

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