A Walk of Perpetual Second Thoughts

A gym is $2$ km from home. Starting from home, a walker covers $\tfrac34$ of the way to the gym, then changes his mind and walks $\tfrac34$ of the way back home, then $\tfrac34$ of the way to the gym again, and so on, always turning around after covering $\tfrac34$ of the distance to his current destination. In the long run he oscillates between a point $A$ km from home and a point $B$ km from home. Find $|A - B|$.