\[\displaystyle \int \limits_0^\infty \int \limits_0^\infty \int \limits_0^\infty \int \limits_0^\infty \frac{1}{wxyz \left(w+x+y+z+\dfrac1w + \dfrac1x + \dfrac1y + \dfrac1z \right)^2} \ dw \ dx \ dy \ dz \]

If the integral above is equal to \(\dfrac{A}{B} \zeta(C)\), where \(A\) and \(B\) are coprime positive integers and \(C\) is an integer, find \(A+B+C\).

**Notation**: \(\zeta(\cdot) \) denotes the Riemann zeta function.

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