Circles centered at $A$ and $B$ each have radius $3$, as shown. Point $O$ is the midpoint of $\overline{AB}$, and $AO=6$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $EF$ is a common tangent. What is the area of the shaded region $ECODF$?