Shourya's non-real roots

Algebra Level 5

For each real number rr, let XrX_r denote the sum of all non-real roots of the equation 128y4128y3+64y216y+r=0128y^4-128y^3+64y^2-16y+r=0. Let SS be the sum of all possible distinct values of XrX_r. Given that SS can be written as ab\frac{a}{b}, where aa and bb are coprime positive integers, what is the value of a+ba+b?

This problem is posed by Shourya P.

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