Determine all sequences $(x_1,x_2,\cdots,x_{2011})$ of positive integers such that for every positive integer $n$ there is an integer $a$ with $$x_1^n+2x_2^n+\cdots+2011x_{2011}^n=a^{n+1}+1.$$ Please provide the value of $x_1 + x_2 + \cdots + x_{2011}$ for the sequence that satisfies the condition.

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