Think Ouside the Box

Geometry Level 3

In rectangle \(ABCD\), \(AB=5\) and \(BC=10\). Points \(E\), \(F\), \(G\), and \(H\) are on \(AD\) and \(BC\) in such a way that \(EF=1\), \(GH=2\), \(AE=3\), and \(BG=6\). \(EG\) and \(FH\) intersect diagonal \(BD\) at points \(Q\) and \(P\), respectively. The length \(PQ\) can be expressed in the form \(\dfrac{x\sqrt{y}}{z}\), where \(x,y\) and \(z\) are positive integers with \(y\) square-free and \(x,z\) coprime. Find \(x+y+z\).

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