HMMT November 2016 T4

Meghal is playing a game with 2016 rounds $1,2, \cdots, 2016$. In round $n$, two rectangular double-sided mirrors are arranged such that they share a common edge and the angle between the faces is $\frac{2 \pi}{n+2}$. Meghal shoots a laser at these mirrors and her score for the round is the number of points on the two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have after she finishes the game?