
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 an integer \(n > 1\), the numbers \(1, 2, 3, \ldots, n\) are written in order on a blackboard. The following moves are possible.
(i) Take three adjacent numbers \(x, y, z\) whose sum is a multiple of 3 and replace them with \(y, z, x\).
(ii) Take two adjacent numbers \(x, y\) whose difference is a multiple of 3 and replace them with \(y, x\).
Find all \(n\) such that the initial list can be transformed into \(n, 1, 2, \ldots, n-1\) after a finite number of moves.

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