# Point in Octagon

Level pending

A point $$P$$ is located in a regular octagon $$ABCDEFGH$$ with side length $$4$$ such that $$\angle GAP=60^{\circ}$$ and $$\angle FGP=\angle BAP$$. If the average of the shortest distance between $$P$$ to the extensions of each face can be expressed as $$a\sqrt{b}+c$$, where $$a,b,c$$ are positive integers and $$b$$ is not divisible by any square, then what is $$abc$$?

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