
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  $a_1+\ldots +a_n$  and  $a_1a_2\ldots a_n$  are coprime.

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