Line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=4$. Three semicircles of radius 2, $AEB$, $BFC$, and $\stackrel{⌢}{\mathrm{CGD}}$, 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 4 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 $a+b+c+d$? The figure is analogous to the one provided in the original problem, with scaled dimensions. (
)