\[ \large \sum_{n=1}^\infty \dfrac{ \zeta(2n)}{(n+1) 2^{4n}} = \dfrac AB - \dfrac {CK}\pi - \dfrac 1D \ln F + \dfrac G{H\pi^I} \zeta(J) \]

The equation above holds true for positive integers \(A,B,C,D,F,G,H,I\) and \(J\) such that \(\gcd(A,B) = \gcd(G,H) = 1 \) and \(F\) is minimized.

Find \(A+B+C+D+F+G+H+I+J \).

**Notations**:

\(\zeta(\cdot) \) denotes the Riemann zeta function.

\(K \) denotes the Catalan's constant, \(\displaystyle K= \sum_{n=0}^\infty \dfrac{ (-1)^n}{(2n+1)^2} \approx 0.916 \)

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