\[S=\frac{1}{4(1)^{4}+1}+\frac{2}{4(2)^{4}+1}+\frac{3}{4(3)^{4}+1}+\dots+\frac{2016}{4(2016)^{4}+1}\]

If the sum above \(S = \dfrac{2017p}{2017q+1}\), where \(p\) and \(q\) are positive integers, find \(p+q\).

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