Let $x,$ $y,$ and $z$ be positive real numbers.  Then the minimum value of
\[\frac{(x^4 + 1)(y^4 + 1)(z^4 + 1)}{xy^2 z}\]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 $a + b + c.$

Let's think step by step and output the final answer within \boxed{}.
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