Partial fraction challenge

Algebra Level 5

S=14(1)4+1+24(2)4+1+34(3)4+1++20164(2016)4+1S=\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=2017p2017q+1S = \dfrac{2017p}{2017q+1}, where pp and qq are positive integers, find p+qp+q.

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