
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 a set \( A = \{a_1, a_2, a_3, a_4\} \) consisting of four distinct positive integers, define \( S_A = a_1 + a_2 + a_3 + a_4 \). Let \( n_A \) be the number of pairs \( (i, j) \) with \( 1 \leq i < j \leq 4 \) such that \( (a_i + a_j) \) divides \( S_A \). Find all sets \( A \) such that \( n_A \) reaches its maximum value.

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