
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
Let \( n \) be a positive integer divisible by 4. Find the number of permutations \(\sigma\) of the set \(\{1, 2, 3, \ldots, n\}\) such that the following conditions are satisfied:
- \(\{\sigma(1), \sigma(2), \ldots, \sigma(n)\}\) is a permutation of \(\{1, 2, \ldots, n\}\), and
- \(\sigma(j) + \sigma^{-1}(j) = n + 1\) for all \( j \in \{1, 2, \ldots, n\} \), where \(\sigma^{-1}(j)\) denotes the position of \( j \) in the permutation \(\sigma\).

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