
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 G be the set of polynomials of the form P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, where c_1,c_2,\cdots, c_{n-1} are integers and P(z) has n distinct roots of the form a+ib with a and b integers. Calculate the number of polynomials in G.

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