
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 all positive integers   $k$  such that for the first  $k$  prime numbers  $2, 3, \ldots, p_k$  there exist positive integers  $a$  and  $n>1$ , such that  $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$ .

*V. Senderov*

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