Two circles of radius $1$ are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,\sqrt{3})$ to $(2,\sqrt{3})$. What is the probability that circles $A$ and $B$ intersect?