\[\sum _{ n=1 }^{ \infty }{ \frac { { \left( { H }_{ n } \right) }^{ 3 } }{ { n }^{ 4 } } } =\frac { A }{ B } \zeta \left( C \right) -\frac { D }{ E } \zeta \left( F \right) \zeta \left( G \right) +H\zeta \left( I \right) \zeta \left( J \right) \]

Here, \(A,B,C,D,E,F,G,H,I,J\) are positive integers.

Find: \(\min { \left\{ A+B+C+D+E+F+G+H+I+J \right\} } \)

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