Decimate the decimals

Algebra Level 4

Let x,y,zx,y,z be positive numbers such that x+y+z=2x+y+z=2 and xy+yz+zx=1xy+yz+zx=1. Given that x20.17+y20.17+z20.17\large x^{20.17}+y^{20.17}+z^{20.17} achieves its maximum value when (x,y,z)=(X,Y,Z)(x,y,z)=(X,Y,Z), and that XYZ=ab,XYZ=\frac{a}{b}, where aa and bb are coprime positive integers, find a+ba+b.

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