Middle European Mathematical Olympiad 2021 Problem I-4

Let $n \geqslant 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n - 1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots, n - 1\}$. Prove that if after $\lceil \frac{n}{2} \rceil$ jumps Zagi has visited $\lceil \frac{n}{2} \rceil + 1$ distinct vertices, then after $n - 1$ jumps Zagi will have visited all of the vertices.

(Remark. For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)