HMMT 2012 G6

Triangle $A B C$ is an equilateral triangle with side length 1 . Let $X_{0}, X_{1}, \ldots$ be an infinite sequence of points such that the following conditions hold:

• $X_{0}$ is the center of $A B C$
• For all $i \geq 0, X_{2 i+1}$ lies on segment $A B$ and $X_{2 i+2}$ lies on segment $A C$.
• For all $i \geq 0, \measuredangle X_{i} X_{i+1} X_{i+2}=90^{\circ}$.
• For all $i \geq 1, X_{i+2}$ lies in triangle $A X_{i} X_{i+1}$.

Find the maximum possible value of $\sum_{i=0}^{\infty}\left|X_{i} X_{i+1}\right|$, where $|P Q|$ is the length of line segment $P Q$.