If \(S\) is a sequence such that

\({ x }_{ n+1 }\quad =\quad \frac { { x }_{ n }+{ x }_{ n }^{ 2 } }{ 1+{ x }_{ n }+{ x }_{ n }^{ 2 } } \)

and \({ x }_{ 1 }=\frac { 1 }{ 2 } \)

then 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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