The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve? $\textbf{(A)}\ 2\pi+6\qquad$ $\textbf{(B)}\ 2\pi+4\sqrt{3}\qquad$ $\textbf{(C)}\ 3\pi+4\qquad$ $\textbf{(D)}\ 2\pi+3\sqrt{3}+2\qquad$ $\textbf{(E)}\ \pi+6\sqrt{3}$