Find minimum of $\frac x {x^2+1} + \frac y{y^2+1} + \frac z{z^2+1}$ Find minimum of $$M=\frac x {x^2+1} + \frac y{y^2+1} +  \frac z{z^2+1}$$
where $x,y,z \in \mathbb R\wedge x+y+z=xy+yz+xz$

I tried:
$$M=\sum \frac 1{x+\frac {1}{x}}\ge \frac 9{\sum{x+\frac 1x}}.$$
$$\text{So, we need to find maximum of } \sum x+ \frac 1x=x+y+z+\frac {x+y+z}{xyz}$$
but how ?

If I understood the question, the minimum seems to be $-\frac{1}{2}$ with $x=1$,$y=-1$ and  $z=-1$.
