
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
Consider the set  $S$  of permutations of  $1, 2, \dots, 2022$  such that for all numbers  $k$  in the
permutation, the number of numbers less than  $k$  that follow  $k$  is even.

For example, for  $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If  $|S| = (a!)^b$  where  $a, b \in \mathbb{N}$ , then find the product  $ab$ .

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