
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 \( 2n + 1 \) cards. There are two cards with each integer from 1 to \( n \) and a joker. The cards are arranged in a line with the joker in the center position (with \( n \) cards on each side of it). For which \( n < 11 \) can we arrange the cards so that the two cards with the number \( k \) have just \( k-1 \) cards between them (for \( k = 1, 2, \ldots, n \))?

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