Determine all sequences \((x_{1},....,x_{2011})\) of positive integers such for every positive integer \(n\) there is an integer a with

\(x_{1}^{n}+2x_{2}^{n}+....+2011x_{2011}^{n}=a^{n+1}+1\).

Find the sum of all of the entries of all of the sequences.

Note: Suppose the solutions are \((1,2,3...,2011)\) and\((2,4,6,8,...,4022)\), then your answer will be \(1+2+3+...+2011+2+4+6+....+4022=6069198\)

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