Three. (20 points) Given the parabola $C: y=\frac{1}{2} x^{2}$, $A_{1}\left(x_{1}, 0\right)$ and $A_{2}\left(x_{2}, 0\right)$ are two points on the $x$-axis $\left(x_{1}+x_{2} \neq 0, x_{1} x_{2} \neq 0\right)$. Perpendicular lines to the $x$-axis are drawn through points $A_{1}$ and $A_{2}$, intersecting the parabola $C$ at points $A_{1}^{\prime}$ and $A_{2}^{\prime}$, respectively. The line $A_{1}^{\prime} A_{2}^{\prime}$ intersects the $x$-axis at point $A_{3}\left(x_{3}, 0\right)$. In this way, $x_{1}$ and $x_{2}$ determine $x_{3}$. Similarly, $x_{2}$ and $x_{3}$ can determine $x_{4}, \cdots$. Given $x_{1}=6, x_{2}=2$, find the value of $x_{6}$.