
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 \( n \) such that there exist \( n \) distinct three-element subsets \( A_{1}, A_{2}, \ldots, A_{n} \) of the set \(\{1,2, \ldots, n\}\) such that \(\left|A_{i} \cap A_{j}\right| \neq 1\) for all \( i \) and \( j \) with \( 1 \leq i < j \leq n \).

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