Excel in math, science, and engineering

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In triangle ABC, if the minimum value of \[\frac { \sum { \cot ^{ 2 }{ \frac { A }{ 2 } } \cot ^{ 2 }{ \frac { B }{ 2 } } } }{ \prod { \cot ^{ 2 }{ \frac { A }{ 2 } } } } \] is equal to 'n', then find 100n

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