jmc

Let $x,$ $y,$ and $z$ be positive real numbers. Then the minimum value of $$\frac{(x^4+1)(y^4+1)(z^4+1)}{x{y}^{2}z}$$is of the form $\frac{a\sqrt{b}}{c},$ for some positive integers $a,$ $b,$ and $c,$ where $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of a prime. Enter $a+b+c.$