
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  $p$  be a prime satisfying  $p^2\mid 2^{p-1}-1$ , and let  $n$  be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer  $N$  such that there exist polynomials  $g(x)$ ,  $h(x)$  with integer coefficients and an integer  $r$  satisfying  $f(x) = (x-r)^N g(x) + p \cdot h(x)$ .

*Proposed by Victor Wang*

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