
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
The polynomial $x^4 - 4008x^3 + ox^2 + px + q$ has integer coefficients with four positive distinct zeros. Exactly one zero is an integer, and is the sum of two other zeros. How many values of $q$ are possible?

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