A closed curve is constructed around a regular hexagon of side length $S=2$. The curve consists of 6 circular arcs, all having radius $R=2$. Three of these arcs are centered at the vertices of an inner equilateral triangle and form outward-curving lobes, each subtending an angle of $\frac{2\pi }{3}$ radians. The remaining three arcs are centered at the vertices of the central hexagon and form inward-curving dips, each subtending an angle of $\frac{\pi }{3}$ radians. Determine the area enclosed by this curve.