
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
Given the set of polynomials $P(z)=z^{n}+c_{n-1} z^{n-1}+c_{n-2} z^{n-2}+\cdots+c_{1} z+50$, where $c_{1}, c_{2}, \cdots, c_{n-1} \in \mathbb{Z}$ and $P(z)$ has distinct complex roots of the form $a+b \mathrm{i}$ $(a, b \in \mathbb{Z})$, determine the number of polynomials in this set.

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