A circle centered at $O$ has radius $2$ and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\angle AOB=30^\circ$. Point $C$ lies on $\overline{OA}$ so that $\overline{BC}$ bisects $\angle ABO$. What is $OC$?

1. $AB$ is tangent to the circle at $A$, so $\angle OAB=90^\circ$. With $OA=2$ and $\angle AOB=30^\circ$: $AB=2\tan30^\circ=\dfrac2{\sqrt3}$ and $OB=\dfrac{2}{\cos30^\circ}=\dfrac4{\sqrt3}$.
2. $BC$ bisects $\angle ABO$, so by the angle bisector theorem in $\triangle OAB$, $\dfrac{OC}{CA}=\dfrac{OB}{AB}=\dfrac{4/\sqrt3}{2/\sqrt3}=2$.
3. Since $OC+CA=OA=2$, we get $OC=\dfrac23\cdot2=\boxed{\dfrac43}$.