Problem 25 (2017 AMC 8)

In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length $2$, and $m\angle TUS = 60^\circ$. Arcs $TR$ and $SR$ are each one-sixth of a circle with radius $2$. What is the area of the region shown? [See figure in original PDF]