CAPS 2015 Problem 3

Real numbers $x,y,z$ satisfy $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0$$ and none of them lies in the open interval $(-1,1)$ . Find the maximum value of $x+y+z$ .

Proposed by Jaromír Šimša