
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

Given a positive integer \( n \), let \( x_{1}, x_{2}, \cdots, x_{2n} \) be a permutation of \( 1, 2, \cdots, 2n \) such that
\[ 
\left| x_{i} - i \right| = 1 \quad (i = 1, 2, \cdots, 2n)
\]
holds for all \( i \). Determine the number of such permutations.

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