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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