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\[a^n-b^{n+1}=(a+b)^{n-1}\]

Given that \(a, b, n\) are positive integers such that \(\gcd(a, b)=1\) and \(n>1,\) find all solutions \((a, b, n)\) to the equation above and enter your answer as \(\sum (a+b+n).\)


Inspiration: [Russian Mathematical Olympiad, 1997]

For prime numbers \(p\) and \(q,\) solve \(p^3-q^5=(p+q)^2.\)

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