
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
We are given  $2n$  natural numbers
\[1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.\]
Find all  $n$  for which these numbers can be arranged in a row such that for each  $k \leq n$ , there are exactly  $k$  numbers between the two numbers  $k$ .

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