On each side of a square of side length $2$, an equilateral triangle of side length $2$ is constructed outwards. The interiors of the square and the triangles have no points in common. Let $R$ be the region formed by the union of the square and these $4$ triangles, and $S$ be the smallest convex polygon that contains $R$. What is the area of the region that is inside $S$ but outside $R$?