
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
Find all positive integers  $n$  such that there exists an infinite set  $A$  of positive integers with the following property: For all pairwise distinct numbers  $a_1, a_2, \ldots , a_n \in A$ , the numbers  $$ a_1 + a_2 + \ldots + a_n \text{ and } a_1\cdot a_2\cdot \ldots\cdot a_n $$  are coprime.

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