
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
Determine all positive integers \( n \) for which there exists an infinite subset \( A \) of the set \( \mathbb{N} \) of positive integers such that for all pairwise distinct \( a_{1}, \ldots, a_{n} \in A \), the numbers \( a_{1} + \cdots + a_{n} \) and \( a_{1} \cdots a_{n} \) are coprime.

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