2022 SSMO Team Round Problems/Problem 13

Problem

In regular hexagon with side length 1, an electron starts at point . When the electron hits an edge, it reflects off of it, with the angle of reflection equal to the angle of incidence. The electron first travels in a straight line to a point on edge . The electron bounces off of a total of edges before hitting a vertex. The electron stops and its total distance traveled is measured. If the shortest possible distance the electron could have traveled can be expressed as , find .