Harvard-MIT Mathematics Tournament

Problem:

Triangle $ABC$ 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 $ABC$.
- For all $i \geq 0$, $X_{2i+1}$ lies on segment $AB$ and $X_{2i+2}$ lies on segment $AC$.
- 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 $\sum_{i=0}^{\infty} |X_{i} X_{i+1}|$, where $|PQ|$ is the length of line segment $PQ$.