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 \(\arctan \left(a\sqrt{\dfrac{b}{c}} \right)\), where \(a,b,c\) are positive integers, and \(b,c\) are relatively prime and square-free. Find \(a+b+c\).

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