Putnam 2018 A4

Let $m$ and $n$ be positive integers with $\gcd(m,n) = 1$, and let $$a_k = \left\lfloor \frac{mk}{n} \right\rfloor - \left\lfloor \frac{m(k-1)}{n} \right\rfloor$$ for $k=1,2,\dots,n$. Suppose that $g$ and $h$ are elements in a group $G$ and that $$gh^{a_1} gh^{a_2} \cdots gh^{a_n} = e,$$ where $e$ is the identity element. Show that $gh= hg$. (As usual, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)