Two circular arcs have the same center $O$. The major arc $\widehat{AD}$ lies on a circle of radius $OA$, and the minor arc $\widehat{BC}$ lies on a larger circle. Point $A$ lies between $O$ and $B$, and point $D$ lies between $O$ and $C$. Each of the segments $\overline{AB}$ and $\overline{CD}$ has length $3$, and major arc $\widehat{AD}$ and minor arc $\widehat{BC}$ each have length $4\pi$. What is the distance from $O$ to $A$?

1. Let $r=OA$ and $\alpha=\angle AOD$ in radians. The major arc $AD$ has central angle $2\pi-\alpha$ and the arc $BC$ has radius $r+3$ and central angle $\alpha$.
2. So $r(2\pi-\alpha)=4\pi$ and $(r+3)\alpha=4\pi$.
3. From the second equation $\alpha=\dfrac{4\pi}{r+3}$. Substituting into the first and dividing by $4\pi$: $r\left(\dfrac12-\dfrac1{r+3}\right)=1$, i.e. $r(r+1)=2(r+3)$, so $r^2-r-6=0$ and $(r-3)(r+2)=0$.
4. Therefore $OA=\boxed{3}$. (Check: $\alpha=\frac{2\pi}3$, the major arc has length $3\cdot\frac{4\pi}3=4\pi$, and arc $BC$ has length $6\cdot\frac{2\pi}3=4\pi$.)