Question 24A plane contains points $A$ and $B$ with $AB = 1.$ Let $S$ be the union of all disks of radius $1$ in the plane that cover $\overline{AB}.$ What is the area of $S?$$2\pi + \sqrt{3}$$\dfrac{8\pi}{3}$$3\pi - \dfrac{\sqrt{3}}{2}$$\dfrac{10\pi}{3} - \sqrt{3}$$4\pi - 2\sqrt{3}$