For any interval $\mathcal{A}$ on the real number line not containing zero, define its reciprocal to be the set of numbers of the form $\frac{1}{x}$ where $x$ is an element in $\mathcal{A}$. Compute the number of ordered pairs of positive integers $(m, n)$ with $m < n$ such that the length of the interval $[m, n]$ is $10^{10}$ times the length of its reciprocal.

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