Henry decides one morning to do a workout, and he walks $\frac{2}{3}$ of the way from his home to his gym. The gym is 2 kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{2}{3}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\frac{2}{3}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\frac{2}{3}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$?