
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 consider permutations \(\left(x_{1}, \ldots, x_{2n}\right)\) of the set \(\{1, \ldots, 2n\}\) such that \(\left|x_{i}-x_{i+1}\right|=n\) for at least one \(i \in\{1, \ldots, 2n-1\}\). For every natural number \(n\), find out whether permutations with this property are more or less numerous than the remaining permutations of \(\{1, \ldots, 2n\}\).

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