Compute expected arc length on a circle

Take the unit circle (radius $1$, centered at the origin) and choose three points independently and uniformly at random on its circumference. The three chosen points split the circle into three circular arcs. Let $L$ be the arc length (measured along the circumference) of the arc that contains the fixed point $(1,0)$ (angle 0). Compute $\mathbb{E}[L]$.

Overview: This problem tests geometric probability, circular order statistics, and computing expectations of continuous random variables. See the full interview experience where this question appeared.