
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
Find integer  $ n$  with  $ 8001 < n < 8200$  such that  $ 2^n \minus{} 1$  divides  $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$  for all integers  $ k > n$ .

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