Three concentric circles centered at $O$ have radii of $1$, $2$, and $4$. Points $B$ and $C$ lie on the largest circle. The innermost circle is shaded, as is the portion of the region between the two larger circles bounded by central angle $\angle BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle BOC$ in degrees?