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 ?

Solution 1:

If $x=y=-1$ and $z=1$ we get a value $-\frac{1}{2}$ and it remains to prove that $$\sum_{cyc}\frac{x}{x^2+1}\geq-\frac{1}{2}$$ or $$(x+y+z+xyz)^2+(x+y+z+1)^2+\sum_{cyc}(x-yz)^2\geq0$$ and we are done!

Solution 2:

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