Eight Circles and a Rhombus

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 $X+Y$ ?