Stunning Summation

Algebra Level 5

\[\sum _{ n=1 }^{ \infty }{ \frac { { 8 }^{ n } }{ \left( 3n-1 \right) ! } } =\frac { A }{ B } { e }^{ -C }\left( { e }^{ D }-E\sin { \left( \frac { \pi }{ F } +\sqrt { G } \right) } \right) \]

If \(A,B,C,D,E,F\) and \(G\) are positive integers satisfying the equation above, find \(\min(A+B+C+D+E+F+G ) \).

Hint: Algebraic approach is easier than calculus approach.

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