Find the sum of the $2007$ roots of 
\[(x-1)^{2007} + 2(x-2)^{2006} + 3(x-3)^{2005} + \cdots + 2006(x-2006)^2 + 2007(x-2007).\]

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