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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