The numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them, and writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?

Let's think step by step and output the final answer within \boxed{}.
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