A circle of radius $R=2$ is centered at the origin. A regular hexagon is inscribed in this circle. A square of side length $2$ is placed such that its bottom-left corner is at $(0,0)$ and its sides are along the positive x and y axes. The shaded area is the area of the circle, minus the area of the hexagon, plus the area of the square. This area 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 $a+b+c+d$?