Question 22
As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\widehat{AEB},$ $\widehat{BFC},$ and $\widehat{CGD},$ have their diameters on $\overline{AD}$, lie in the same halfplane determined by line $AD$, and are tangent to line $EG$ at $E,$ $F,$ and $G,$ respectively. A circle of radius $2$ has its center at $F.$ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form $$
\frac{a}{b}\cdot\pi-\sqrt{c}+d,
$$ where $a,$ $b,$ $c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d?$
\frac{a}{b}\cdot\pi-\sqrt{c}+d,
$$ where $a,$ $b,$ $c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d?$