2020 HMMT November General Round Problem 9

In the Cartesian plane, a perfectly reflective semicircular room is bounded by the upper half of the unit circle centered at (0,0) and the line segment from (-1,0) to (1,0). David stands at the point (-1,0) and shines a flashlight into the room at an angle of 46^\circ above the horizontal. How many times does the light beam reflect off the walls before coming back to David at (-1,0) for the first time?

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2020 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.