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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