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Calculus Level 5

\[\lim _{x\rightarrow2}{\left(\frac { \pi \ln { \left| 2\sin { \pi x } \right| } }{ 2\sinh ^{ 2 }{ \pi x } } -\frac { \pi \ln { \left| 16-{ x }^{ 4 } \right| } }{ 2\sinh ^{ 2 }{ 2\pi } }\right) } = \frac { A\pi \ln { B\pi } -\pi \ln { C } }{ D\sinh ^{ E }{ F\pi } } \]

If the equation above holds true for positive integers \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\), find \(A+B+C+D+E+F\).

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