Super Cool Geometry

If perpendiculars be drawn on the sides of a regular polygon of $n$ sides from any point on the inscribed circle whose radius is $a$ , then the following conditions hold true :-

$\frac{2}{n} \times \sum (\frac{p}{a})^{2} = b$ and $\frac{2}{n} \times \sum (\frac{p}{a})^{3} = c$
- where $p$ is the length of the perpendicular

Find $b^{7} + c^{11}$

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