<s>[INST] Minimum value of: $x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$ 
$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:
  $$x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$$

I put it in the form $x^6y +x^6z+y^6x+y^6z+z^6x +z^6y$. I tried AM-GM but it's not helping.
 [/INST] Step:$1$ $$x^6y +x^6z+y^6x+y^6z+z^6x +z^6y \geq 6x^2y^2z^2(xyz)^{\frac{1}{3}}$$
Step:$2$$$\frac{\frac{1}{yz}+\frac{1}{xz}+\frac{1}{xy}}{3} \geq \frac{1}{(xyz)^{\frac{2}{3}}}$$
$$(xyz)^{\frac{2}{3}} \geq 3$$ 
Next step : Substitute result of step:$2$ in step:$1$
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