Points, Lines & Planes · Superchallenge problem

Let $A = (0, 0)$, $B = (14, 0)$, and $C = (0, 7)$. A token starts at some point of the plane and moves repeatedly: first to the midpoint of its current position and $A$, then to the midpoint of its new position and $B$, then to the midpoint of its position and $C$, then toward $A$ again, and so on, cycling $A, B, C, A, B, C, \ldots$. Whatever the starting point, the positions reached just after moving toward $A$ get closer and closer to a point $X$, and the positions reached just after moving toward $B$ get closer and closer to a point $Y$. Find the distance $XY$. Give your answer in simplest radical form.