An ant travels along a straight path with a distance of meters. From then on, it turns left and covers of the straight distance it traveled before turning, and continues doing this until it eventually reaches an unknown point (referred to as the red point in the middle of the picture). Refer to the figure above.
The distance of from the ant's starting point is meters, where is a positive constant which can be expressed in the form , where and is square free. Determine the value of .
Challenge: Determine an explicit formula for .