Points on a curve

On the $xy$ -coordinate plane, $A_{0}=(0,0), A_{1}, A_{2}, A_{3}\cdots$ ... are points that lie on the $x$ -axis and $B_{1}, B_{2}, B_{3}, B_{4}\cdots$ are points that lie on the curve $y=\sqrt{x}$ such that for all natural numbers $k$ , the triangle formed by $A_{k-1}$ , $A_{k}$ and $B_{k}$ is equilateral. Joel randomly picks a natural number $l$ . What is the probability that $A_{l}$ has integer coordinates? Give your answer to 3 decimal places.