
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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Let $H$ 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 that are either pure integers or purely imaginary (of the form $bi$ where $b$ is an integer). How many such polynomials are in $H$?

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