The minimum value of $\dfrac{(x^4+1)(y^4+1)(z^4+1)}{xy^2z}$ as $x,y,$ and $z$ range over the positive reals is equal to $\dfrac{A\sqrt{B}}{C},$ where $A$ and $C$ are coprime and $B$ is squarefree. What is $A+B+C?$

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