
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
A deck of  $ 2n\plus{}1$  cards consists of a joker and, for each number between 1 and  $ n$  inclusive, two cards marked with that number. The  $ 2n\plus{}1$  cards are placed in a row, with the joker in the middle. For each  $ k$  with  $ 1 \leq k \leq n,$  the two cards numbered  $ k$  have exactly  $ k\minus{}1$  cards between them. Determine all the values of  $ n$  not exceeding 10 for which this arrangement is possible. For which values of  $ n$  is it impossible?

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