
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
assume that X is a set of n number.and  $0\leq k\leq n$ .the maximum number of permutation which acting on  $X$  st every two of them have at least k component in common,is  $a_{n,k}$ .and the maximum nuber of permutation st every two of them have at most k component in common,is  $b_{n,k}$ .
a)proeve that : $a_{n,k}\cdot b_{n,k-1}\leq n!$ 
b)assume that p is prime number,determine the exact value of  $a_{p,2}$ .

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