Question 15
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 $1,$ $AEB,$ $BFC,$ and $CGD,$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,$ $F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F.$ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
$$
\dfrac{a}{b}\cdot\pi-\sqrt{c}+d,
$$
where $a,$ $b,$ $c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d?$
$$
\dfrac{a}{b}\cdot\pi-\sqrt{c}+d,
$$
where $a,$ $b,$ $c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d?$