
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 $$ Q(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+75, $$where $ c_1,c_2,\dots, c_{n-1} $ are integers and $Q(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?

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