Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property that for every $x, y \in A$ with $x \neq y$, it holds that \[\left| \frac{1}{x} - \frac{1}{y} \right| > \frac{1}{1000}.\] Determine the largest possible number of elements that the set $A$ can have.

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