Circles centered at $A$ and $B$ each have radius $4$, as shown. Point $O$ is the midpoint of $\overline{AB}$. Segment $OC$ is tangent to the circle centered at $A$ at point $C$, and $OC=4$. Segment $OD$ is tangent to the circle centered at $B$ at point $D$, and $EF$ is a common tangent. What is the area of the shaded region $ECODF$?