# Think Ouside the Box

Geometry Level 4

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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