As shown, a solid is composed of cone $P$-$O$ and cone $Q$-$O$. $AB$ is the diameter of circle $O$. $AB=4$, $PA=QB=3$, $C$, $D$ are points on the lines $PA$ and $QB$ respectively, $PC=QD=\sqrt{3}$. An ant crawls on the surface of the solid, starting at C and ending at D. Find the shortest distance the ant can travel.
