On each side of a regular hexagon of side length $1$, an equilateral triangle of side length $1$ is constructed outwards. The interiors of the hexagon and the triangles have no points in common. Let $R$ be the region formed by the union of the hexagon and these $6$ triangles, and $S$ be the smallest regular hexagon that contains $R$. What is the area of the region that is inside $S$ but outside $R$?