What are the min and max values of:

$\displaystyle \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}$

for reals $\displaystyle x,y,z$ such that $\displaystyle xyz=1$?

EDIT: I now see that $\displaystyle 1$ is the only possible value if only xyz=1, but I don't know how to reduce it to show that fact. Anyone?