Let \((a_1,b_1)\), \((a_2,b_2)\), \(...\), \((a_n,b_n)\) be all the pairs of prime numbers that satisfy \[\frac{a}{a+1}+\frac{b+1}{b}=\frac{2n}{n+2}\] for some positive integer \(n\). Then find the sum of all possible values of \(b-a\).

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