HMMT November 2019 T9

Will stands at a point $P$ on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of $n^{\circ}$ and $(n+1)^{\circ}$ with the tangent at $P$, where $n$ is a positive integer less than 90 . The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach $P$ again. ( $P$ is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room?