Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$.  Compute the sum
\[\frac{x^2}{x-1} + \frac{x^4}{x^2-1} + \frac{x^6}{x^3-1} + \dots + \frac{x^{4020}}{x^{2010}-1}.\]

Let's think step by step and output the final answer within \boxed{}.
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