Proving the AGM inequality with 3 variables
Because of you've done with four variables, you can use it like this.
$$x+y+z + \frac{x+y+z}{3} \geq 4\sqrt[4]{xyz(\frac{x+y+z}{3})}$$
then $$\frac{4}{3}(x+y+z) \geq 4\sqrt[4]{xyz(\frac{x+y+z}{3})}$$
or, $$\frac{x+y+z}{3} \geq \sqrt[4]{xyz(\frac{x+y+z}{3})},$$
$$\left(\frac{x+y+z}{3}\right)^3 \geq xyz.$$
With $w = (xyz)^{1/3}$, applying AM GM inequality for four variables, we have $$\frac{x+y+z+w}{4} \geq (xyz(xyz)^{1/3})^{1/4} = (xyz)^{1/3}$$ The required inequality follows.