Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot south, then $1$ foot west, then $1$ foot down, then $1$ foot north, and then continues to repeat this pattern. Bee $B$ travels $1$ foot east, then $1$ foot up, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly $13$ feet away from each other?