
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 $x^4 - 5020x^3 + mx^2 + nx + p$ with integer coefficients. This polynomial has four distinct positive zeroes. Exactly one of these is an integer, and it is the sum of the other three zeroes. How many values of $p$ are possible?

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