
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
Consider the polynomial $x^4 - 2004x^3 + mx^2 + nx + p$ that has integer coefficients and four distinct positive zeros. Exactly one zero is an integer, which is the sum of two of the other zeros, and the fourth zero is the product of the other three. Determine how many values of $p$ are possible.

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