Let $\omega =-\frac{1}{2}+\frac{1}{2}i\sqrt{3}$. Let $S$ denote all points in the complex plane of the form $a+b\omega +c{\omega }^2$, where $0\le a\le 2$, $0\le b\le 1$, and $0\le c\le 1$. What is the area of $S$?