Nemotron-IMO-Bench 2026 A43

Let $p$ be an odd prime number, and let $\mathbb{F}_p[X]$ be the ring of polynomials over the field $\mathbb{F}_p$ of integers modulo $p$. Let $q\in \mathbb{F}_p[X]$ be given by $$q(X) = \sum_{k=1}^{p-1} k^{\frac{p-1}{2}} X^k.$$ Find the greatest nonnegative integer $n$ such that $(X-1)^n$ divides $q$ in $\mathbb{F}_p[X]$.