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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