# Eight Circles and a Rhombus

Geometry Level 5

The figure above is rhombus $$PQRS$$ with a common side length of $$10$$. Also shown above are circles $$A$$ and $$A'$$, each with radius $$10$$. Both circles have the side $$\overline{PS}$$ as a chord.

Similarly (but not shown) are $$6$$ more circles $$B$$, $$B'$$, $$C$$, $$C'$$, $$D$$, and $$D'$$, each with a radius of $$10$$, where $$\overline{RS}$$ is a chord of $$B$$ and $$B'$$, $$\overline{QR}$$ is a chord of $$C$$ and $$C'$$, and $$\overline{PQ}$$ is a chord of $$D$$ and $$D'$$.

Finally, $$A$$, $$B$$, $$C$$, and $$D$$ are connected to form quadrilateral $$ABCD$$, and the centers of $$A'$$, $$B'$$, $$C'$$, and $$D'$$ are connected to form quadrilateral $$A'B'C'D'$$.

If the area of $$PQRS$$ is $$96$$, the area of $$ABCD$$ is $$X$$, and the area of $$A'B'C'D'$$ is $$Y$$, what is $$X+Y$$?

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