Maximum of a Big Fraction

Algebra Level 5

x,yx, y and zz are positive reals. What is the maximum value of

((x+y+z)3x3y3z3)2(x2+y2+z2)3x6y6z6? \frac { \big( (x+y+z)^3 - x^3-y^3-z^3 \big)^2} {(x^2+y^2+z^2)^3 -x^6-y^6-z^6} ?

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