Welcome 2016! Part 23

Let \(a\) and \(b\) be positive integers satisfying \( \dfrac{ab+1}{a+b} < \dfrac{3}{2}\).

The maximum possible value of \( \dfrac{a^3b^3+1}{a^3+b^3}\) is \(\dfrac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Find \(p+q\).

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