\[\large \displaystyle \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \left[\frac{\tanh(\frac{w-x}{2})\tanh(\frac{x-y}{2})\tanh(\frac{y-z}{2})}{\cosh w +\cosh x + \cosh y + \cosh z}\right]^2 \, dw \; dx \; dy \; dz \]

The above integral can be written as \[\large \frac{A}{B}\pi^C - D - E\zeta(F),\]

where \(A,B,C,D,E\) and \(F\) are positive integers with \(A,B\) coprime. Calculate \(A+B+C+D+E+F\).

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

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