$3\geq\sum\limits_{cyc}\frac{(x+y)^{2}x^{2}}{(x^{2}+y^{2})^{2}}$ with $x,y,z >0$

Let $x,y,z>0$. Prove that: $$3\geq \frac{(x+ y)^{2}x^{2}}{(x^{2}+ y^{2})^{2}}+ \frac{(y+ z)^{2}y^{2}}{(y^{2}+ z^{2})^{2}}+ \frac{(z+ x)^{2}z^{2}}{(z^{2}+ x^{2})^{2}}$$

I need to the hints and hope to see the Buffalo Way help here! Thanks a lot!

My idea is as follows: Because this inequality is cyclic. So, it's enough to prove this inequality in two cases: $x\leq y\leq z$ and $x\geq y\geq z$. I can prove it with $x\leq y\leq z$ but with $x\geq y\geq z$, I can't.

from Hot Weekly Questions - Mathematics Stack Exchange