
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  $(1,2,3,\cdots,n)$  which satisfy the condition  $\sigma(j)+\sigma^{-1}(j)=n+1$  for all  $j \in \{1,2,3,\cdots,n\}$ .

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