
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
Let  $n\ge 2$  be a given integer. Find the greatest value of  $N$ , for which the following is true: there are infinitely many ways to find  $N$  consecutive integers such that none of them has a divisor greater than  $1$  that is a perfect  $n^{\mathrm{th}}$  power.

*Proposed by Péter Pál Pach, Budapest*

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