Winter Mathematical Competition

Let $ABC$ be a right triangle with $\angle ACB=90^{\circ }$, $AC=1$ and $BC=2$. Given a point $A_1\in BC$ such that $A_{1}C\ne \frac{1}{3}$ we construct a sequence of points $A_n\in BC$, $n\ge 2$ in the following way. Let $B_1$ be the intersection point of $AC$ and the line through $A_1$ and parallel to $AB$, and $C_1$ be the foot of the perpendicular from $B_1$ to $AB$. Then $A_2$ is the intersection point of $BC$ and the line through $C_1$ and parallel to $AC$. Using $A_2$ we construct $A_3$ in the same way, etc. Find: a) $\frac{3A_{2}C-1}{3A_{1}C-1}$; b) ${\lim }_{n\to \infty }{S}_{A_n{B}_n{C}_n}$.