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Two nonnegative integers $a$ and $b$ are friends if the decimal expression of $a+b$ is formed only by 0's and I's. Suppose that $A$ and $B$ are two infinite sets of nonnegative integers such that $B$ is the set of all numbers which are friends of all the elements of $A, A$ is the set of all numbers which are friends of all the elements of $B$. Show that one of the sets $A, B$ contains infinitely many pairs of numbers $x, y$ with $x-y=1$.
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