
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
In a banquet, \( n \) guests are invited and they are seated around a circular table with seats numbered \( 1, 2, 3, \ldots, n \). A waiter serves the guests following a peculiar rule: after serving a guest, he moves counterclockwise by a number of seats equal to the number on the card of the guest he just served and serves the guest at the new seat. He continues in this manner, moving counterclockwise by the number on the card of the guest he just served.

Find all positive integers \( n \geq 2 \) for which the host can arrange the \( n \) cards such that the waiter, by starting at an appropriately chosen guest, is able to serve every guest exactly once by following the above rule.
(2003 Italian Math Olympiad)

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