Is it just AM-GM?

Algebra Level 4

Given that x1+x+y1+y+z1+z=2,\dfrac{x}{1+x}+\dfrac{y}{1+y}+\dfrac{z}{1+z}=2, where x,y,zx,y,z are non-negative real numbers, what is the minimal value of xyzxyz?

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