A regular \(n\) sided polygon has vertices \(V_1, \ V_2, \ \cdots \ , \ V_n\).

If \[\dfrac{\overline{V_1 V_2}+\overline{V_1 V_3}+ \cdots \cdots +\overline{V_1 V_7}}{\csc {\dfrac{\pi}{n}}}=\overline{V_1 V_2}\left(\dfrac{1+\cot {\dfrac{\pi}{24}}}{2}\right)\] then find the value of \(n\)

\(\textbf{Assumptions and Source}\)

\(\bullet \ \ \ \ \) \( \overline{V_1 V_i}\) denotes the distance between the first vertex and the \(i^{\text{th}}\) vertex, where \(V_1\) is adjacent to \(V_2\), which is adjacent to \(V_3\) and so on.

\(\bullet \ \ \ \ \) A similar (but easier) question appeared in the \(1994 \ \text{IITJEE exam}\).

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