Grazing Through A Hexagon

Geometry Level 5

A regular hexagon stands with one side on the ground and a particle is projected so as just to graze its four upper vertices. If the angle of projection can be written as arctan(abc)\arctan \left(a\sqrt{\dfrac{b}{c}} \right), where a,b,ca,b,c are positive integers, and b,cb,c are relatively prime and square-free. Find a+b+ca+b+c.

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