
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
Suppose we have a pack of  $2n$  cards, in the order  $1, 2, . . . , 2n$ . A perfect shuffle of these cards changes the order to  $n+1, 1, n+2, 2, . . ., n- 1, 2n, n$  ; i.e., the cards originally in the first  $n$  positions have been moved to the places  $2, 4, . . . , 2n$ , while the remaining  $n$  cards, in their original order, fill the odd positions  $1, 3, . . . , 2n - 1.$ Suppose we start with the cards in the above order  $1, 2, . . . , 2n$  and then successively apply perfect shuffles. 
What conditions on the number  $n$  are necessary for the cards eventually to return to their original order? Justify your answer.

<details><summary>Remark</summary>Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.</details>

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