Middle European Mathematical Olympiad 2024 Problem I-2

There is a sheet of paper (like this one) on an infinite blackboard. Marvin secretly chooses a convex 2024-gon $P$ that lies fully on the piece of paper. Tigerin wants to find the vertices of $P$. In each step, Tigerin can draw a line $g$ on the blackboard that is fully outside the piece of paper, then Marvin replies with the line $h$ parallel to $g$ that is the closest to $g$ which passes through at least one vertex of $P$. Prove that there exists a positive integer $n$ such that Tigerin can always determine the vertices of $P$ in at most $n$ steps.