KöMaL 2011 B4363

The reciprocals of the natural numbers 2 to 2011 are written on a blackboard. In each step, two numbers $x$ and $y$ are erased and replaced with the number $\frac{xy}{xy + (1-x)(1-y)}$. This step is repeated 2009 times, until a single number remains. What may the remaining number be? (Suggested by B. Kovács, Szatmárnémeti)