
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 $x^4 - 5020x^3 + px^2 + qx + r$ has integer coefficients and four distinct positive zeros. Exactly two of these zeros are integers, each being the sum of the other two zeros (pairwise). Calculate how many values of $r$ are possible?

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