
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 \( n \geq 2 \) such that for any positive real numbers \( a_{1}, a_{2}, \ldots, a_{n} \) satisfying \( \prod_{i=1}^{n} a_{i} = 1 \), the following inequality always holds:
\[ \sum_{i=1}^{n} \frac{a_{i}}{a_{i+1}} + \sum_{i=1}^{n} \frac{a_{i+1}}{a_{i}} \geq 2 \sum_{i=1}^{n} a_{i} \]
where \( a_{n+1} = a_{n} \).

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