Bouncing Laser Beam

Geometry Level 5

You are at one vertex of an equilateral triangle with side length 1. All of the edges of the equilateral triangle will reflect the laser beam perfectly (angle of incidence is equal to angle of reflection). Given that the laser beam bounces off exactly 137 edges and returns to the original vertex without touching any other vertices, let \(M\) be the maximum possible distance the beam could have traveled, and \(m\) be the minimum possible distance the beam could have traveled. Find \(M^2-m^2\).

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