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$S =\frac{1}{1^{4}+1^{2}+1}+\frac{2}{2^{4}+2^{2}+1}+\frac{3}{3^{4}+3^{2}+1}+\dots+\frac{2016}{2016^{4}+2016^{2}+1}$

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

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