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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