
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  $n \geq 3$  for which there are  $n$  positive integers  $a_1, \cdots , a_n$  any two of which have a common divisor greater than  $1$ , but any three of which are coprime. Assuming that, moreover, the numbers  $a_i$  are less than  $5000$ , find the greatest possible  $n$ .

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