
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
For \( n (n \geq 3) \) integers \( a_{1}, a_{2}, \ldots, a_{n} \), arrange these numbers in a circle. In each operation, take the difference of each pair of adjacent numbers and take the absolute value, resulting in \( n \) new integers, which are then arranged in a circle in the same order. Find all positive integers \( n \) such that for any \( n \) integers, after a finite number of operations, all resulting numbers are the same.

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