Let \(x, y, z\) be positive real numbers such that \(x^2+y^2+z^2=2(xy+yz+zx)\). FInd the maximum value of \[\frac { { x }^{ 10 }+{ y }^{ 10 }+{ z }^{ 10 } }{ (x+y+z)\big({ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }\big)\big({ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }\big)\big({ x }^{ 4 }+{ y }^{ 4 }+{ z }^{ 4 }\big) }\]
to 4 decimal places.

Inspiration.