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If $\displaystyle \sum_{r=1}^\infty \dfrac r{r^4+r^2+1} = 1 - \dfrac1a$, find the number of digits in $a^{2016}$.

You are given the following approximations: $\log 10 = 1, \log 5 = 0.6990, \log 3 = 0.4771$.

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