How many bounces can you find?

Geometry Level pending

Suppose you're bouncing a ball around a unit equilateral triangle, starting on one of the sides. The angle of incidence equals the angle of reflection, and the ball cannot hit any vertices.

Let's call a cycle any way of bouncing the ball such that after a finite number of bounces it reaches the starting point. A cycle ends when it reaches the starting point.

Given that the total length of the cycle is less than or equal to \(2\sqrt { 3 } \) (that is, that total distance it travels before reaching the starting location), what is the maximum number of bounces in the cycle?

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