Great and tough Inequality

Level pending

Let x,y,z be real numbers in \[]-1,1[ \] such that : \[ x+y+z=0 \]

Find the maximum value of :

\[ \sqrt{x+1+\frac{y^2}{6}}+\sqrt{y+1+\frac{z^2}{6}}+\sqrt{z+1+\frac{x^2}{6}} \]

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