\[{ S }_{ n }=\sum _{ i=1 }^{ n }{ \cot ^{ -1 }{ \left( { i }^{ 2 }+i+1 \right) } } \]

Conside the summation above, \(S_n\). If the value of \({ S }_{ 100 }\) can be written as \(\cot ^{ -1 }{ \left( \frac { a }{ b } \right) } \) for coprime positive integers \(a\) and \(b\) with \(a>b\), find the value of \(2(a+b)\).

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