
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
Denote by  $a \bmod b$  the remainder of the euclidean division of  $a$  by  $b$ . Determine all pairs of positive integers  $(a,p)$  such that  $p$  is prime and \[ a \bmod p + a\bmod 2p + a\bmod 3p + a\bmod 4p = a + p. \]

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