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Algebra Level 5

\[x_{n+1}=\frac{x_n+x_n^2}{1+x_n+x_n^2};x_1=\frac{1}{2}\] If {\(x_n\)} is a sequence of numbers such that they satisfy the recurrence relation above, find \[\frac{1}{x_1+1}+\frac{1}{x_2+1}+\frac{1}{x_3+1}+......\frac{1}{x_{2012}+1}+\frac{1}{x_{2013}}\]

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