
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
The numbers $1,2,\cdots,n$ have $n!$ different permutations $a_1,a_2,\cdots,a_n (n\geqslant 2,n\in\mathbb{N}^*)$, among which the number of distinct permutations satisfying "for all $k=1,2,\cdots,n$, $a_k\geqslant k-2$" is ______.

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