Another Max-min problem!

Algebra Level pending

Given two positive real numbers \(x,y\) that satisfy: \(x \ge 1, y \ge 1, 3(x+y)=4xy\).

Let \(P=x^3+y^3+3\left(\displaystyle\frac{1}{x^2}+\displaystyle\frac{1}{y^2}\right)\)

If \(\max {P} + \min {P} = \displaystyle\frac{a}{b}\), where a and b are coprime positive integers, find \(a+b\).

This problem is not original.

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