A Recursive Angle Chase

Suppose that $X_1$ is the point on side $BC$ of $\triangle ABC$ such that $\angle AX_1C = 120^\circ$. Moreover, suppose that $Y_1$ is the point on side $AC$ such that $\angle X_1Y_1C = 120^\circ$. Define the sequences of points $\{X_i\}_{i=1}^\infty$ and $\{Y_i\}_{i=1}^\infty$ such that $X_i$ is on side $BC$ and $Y_i$ is on side $AC$, and where $\angle Y_iX_{i+1}C = \angle X_iY_iC = 120^\circ$ for all $i\in \mathbb{N}$. If $X_1A:X_1B:X_1C = 1:3:1$ and $AY_6 = \dfrac{728}{81}$, then the length of $AB$ can be written in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is square-free. What is $a+b$?