\[\displaystyle \int_{0}^{-\infty} \left [ 2^{at} \zeta(bt) - 2^{c-dt} \zeta(t) \right ] \ dt\]

If \(S = \log_{2}{e} - \log_{4}{e} + \log_{6}{e} - \log_{8}{e} \ldots\) can be expressed as the integral above for positive integers \(a,b,c,d\) are positive integers, find the value of \(200(a^2 + b^2 + c^2 - \sqrt{8d})\).

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