
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  $p$  and  $q$  be positive integers such that  $p$  is a prime,  $p$  divides  $q-1$ , and  $p+q$  divides  $p^2+2020q^2$ . Find the sum of the possible values of  $p$ .

*Proposed by Andy Xu*

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