
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 $H$ be the set of polynomials of the form $$ P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+36, $$ where $ c_1,c_2,\dots, c_{n-1} $ are integers and $P(z)$ has distinct roots including both real integers and purely imaginary numbers. How many polynomials are in $H$?

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