As shown in the diagram, one end of a $11$-foot-long electric cable is fixed at point $A$ on the wall, and the other end of the cable (denoted by $C$) is a sensor. There is a pin attached to the cable at a point $B$ that is $4$ units away from $C$, and the pin can be fixed to any point on the wall reachable by $B$. Due to gravity, the free part of the cable (i.e. $BC$) is always perpendicular to the ground $\overline{OD}.$ A few other measurements are provided in the diagram. What is the minimum distance between $C$ and $D?$
