\[ A = \lim_{ n \to \infty } \frac{ 3 \displaystyle \sum_{ r = 1 }^{ n - 1 } \cot^{ 2 } \left( \frac{ r \pi }{ n } \right) }{ ( n + 1) ( n + 2 ) } \]

Find the value of the following expression if \( \omega = \dfrac{ \pi } { 27 } \).

\[ \tan \left(\frac{ ω }{ 2 } \right) + \sum_{ n = 1 }^{ A + 1} \sin( 3^{ n - 1 } ω ) \sec( 3^{ n }ω ) \]

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