Don't cancel AB

\[\large \dfrac{1}{a} + \dfrac{1}{b} + 1 = \overline{ab} \div (a\times b) \]

For single digit positive integers \(a\) and \(b\), let \(\overline{ab} \) denote the two-digit integer: \(10a + b\).

Suppose \(S_a\) and \(S_b\) be the sum of all possible values of \(a\) and \(b\) respectively such that they satisfy the equation above. Find the value of \(S_a+S_b\).

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