Beautiful But So Dangerous

Algebra Level 4

$\small \frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$

Over all positive reals $x, y$ and $z$ , the maximum of the expression above can be expressed as $\dfrac{m}{n}$ , where $m$ and $n$ are coprime positive integers. Find $m+n$ .

Beautiful But So Dangerous

Source: VMO 2014 problem 2 day 2.