Extremas of a Constrained Inequality!

Algebra Level 5

$\large{ \dfrac{x^4 + y^4 + z^4}{(x+y+z)^4} }$

Let the minimum and maximum values of the above expression be ${\alpha}$ and ${\beta},$ respectively, satisfying the following conditions:

$x,y,z \in \mathbb R^+$
$(x+y+z)^3 = 32xyz.$

Suppose ${\alpha}$ can be represented as $\dfrac{A - B \sqrt{C}}{D},$ and ${\beta}$ as $\dfrac{E}{F},$ for positive integers $A,B,C,D,E,F$ and with $C$ having no square factors.

Then find the minimum value of $A+B+C+D+E+F$ .