Why do We Fall? To Rise Again!

\(a\) and \(b\) are positive integers such that \(a \ge b\) and the expression below is an integer: \[\frac{a+1}{b}+\frac{b+1}{a}.\] Find the sum of all values of \(a\) less than \(1000\) that satisfy the above conditions.


Inspiration: [Problem #1, Spain Mathematical Olympiad, 1996]

\(a\) and \(b\) are positive integers such that \[\frac{a+1}{b}+\frac{b+1}{a}\] is an integer. Show that the greatest common divisor of \(a\) and \(b\) is not greater than \(\sqrt{a+b}\).

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