Find The Maximum

Algebra Level 4

As \(x,y,z\) range over all positive real numbers satisfying \(xy+yz+zx= xyz(x+y+z),\) the maximum value of \[\dfrac{1}{(2x+y+z)^2} + \dfrac{1}{(x+2y+z)^2} + \dfrac{1}{(x+y+2z)^2}\] can be expressed as \(\dfrac{a}{b},\)where \(a,b\) are coprime positive integers. Find \(a+b.\)

Details and assumptions