\[\large \displaystyle \sum_{k=2}^{\infty} \frac{\zeta(k)-1}{k+1} = \frac{A}{B} - \frac{C}{B}\ln(D \pi) - \frac{\gamma}{B}\]

The above equation holds true for positive integers \(A\), \(B\), \(C\), and \(D\). Find \(A+B+C+D\).

**Notation**: \(\gamma \approx 0.5772 \) denotes the Euler-Mascheroni constant.

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