
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
For any set \( A = \{ a_1, a_2, a_3, a_4 \} \) of four distinct positive integers with sum \( s_A = a_1 + a_2 + a_3 + a_4 \), let \( p_A \) denote the number of pairs \( (i, j) \) with \( 1 \leq i < j \leq 4 \) for which \( a_i + a_j \) divides \( s_A \). Among all sets of four distinct positive integers, determine those sets \( A \) for which \( p_A \) is maximal.

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