HMMT 2011 U29

Let $A B C$ be a triangle such that $A B=A C=182$ and $B C=140$. Let $X_{1}$ lie on $A C$ such that $C X_{1}=130$. Let the line through $X_{1}$ perpendicular to $B X_{1}$ at $X_{1}$ meet $A B$ at $X_{2}$. Define $X_{2}, X_{3}, \ldots$, as follows: for $n$ odd and $n \geq 1$, let $X_{n+1}$ be the intersection of $A B$ with the perpendicular to $X_{n-1} X_{n}$ through $X_{n}$; for $n$ even and $n \geq 2$, let $X_{n+1}$ be the intersection of $A C$ with the perpendicular to $X_{n-1} X_{n}$ through $X_{n}$. Find $B X_{1}+X_{1} X_{2}+X_{2} X_{3}+\ldots$