
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
There are \(2n\) cards initially in the order \(1, 2, \cdots, 2n\). The cards are rearranged so that the sequence becomes \(n+1, 1, n+2, 2, \cdots, n+i, i, \cdots, 2n, n\). Specifically, the first \(n\) cards are moved to the positions \(2, 4, \cdots, 2n\) and the last \(n\) cards are placed in the positions \(1, 3, \cdots, 2n-1\). This is called one shuffle. For which values of \(n\), can the sequence \(1, 2, \cdots, 2n\) be restored to its original order after several shuffles?

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