Shourya's non-real roots

Algebra Level 5

For each real number \(r\), let \(X_r\) denote the sum of all non-real roots of the equation \(128y^4-128y^3+64y^2-16y+r=0\). Let \(S\) be the sum of all possible distinct values of \(X_r\). Given that \(S\) can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?

This problem is posed by Shourya P.

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