
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 possible positive integers \( n \geqslant 3 \) such that there exist \( n \) distinct positive real numbers \( a_1, a_2, \cdots, a_n \) for which the products \( a_i a_j \) (where \( 1 \leqslant i < j \leqslant n \)) can be rearranged in increasing order to form a geometric sequence.

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