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 $1$, $\overparen{AEB}$, $\overparen{BFC}$, and $\overparen{CGD}$ 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 $$\frac{a}{b}\cdot\pi -\sqrt{c}+d$$

where $a$, $b$, $c$, and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$