
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
A set \( A = \{a_1, a_2, a_3, a_4\} \) consists of four distinct positive integers. Let the sum \( a_1 + a_2 + a_3 + a_4 \) be denoted as \( s_A \), and let \( n_A \) be the number of pairs \((i, j)\) (where \( 1 \leq i < j \leq 4 \)) such that \( a_i + a_j \) divides \( s_A \).

Determine all sets \( A \) for which \( n_A \) attains the maximum possible value.

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