Question 13
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\{\tfrac{1}{z} \mid z \in R\right\}.$ Then the area of $S$ has the form $a\pi + \sqrt{b},$ where $a$ and $b$ are positive integers. Find $a + b.$