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Let $a_1, a_2, \ldots, a_n$ be positive numbers. Consider the function
$f(x)=\frac{a_1}{x+a_1}+\frac{a_2}{x+a_2}+\cdots+\frac{a_n}{x+a_n} .$
Determine the sum of the lengths of the intervals where $f(x) \geq 1$.
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