\[\Large{A=\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n+1 } } \sum _{ k=0 }^{ n }{ \left( \begin{matrix} n \\ k \end{matrix} \right) } \frac { { \left( -1 \right) }^{ k } }{ k+1 } \\ B=\sum _{ n=2 }^{ \infty }{ \frac { \zeta \left( n \right) -1 }{ n } { Im }\left( { \left[ 1+i \right] }^{ n }-\left[ 1+{ i }^{ n } \right] \right) } }\]

If \(A\) can be expressed as

\[\Large{\frac{\pi^{a}}{b}}\]

And \(B\) can be expressed as

\[\Large{\frac{\pi^{c}}{d} } \].

Find \( (a+b)(c+d)\times \frac{1}{2} \)

**Details and Assumption**:

\(\text{Im}\) represents the Imaginary part of a complex number

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