Strange point in a strange polygon

Geometry Level 5

We have a regular polygon \(A\) with vertices \(A_0, A_1, ..., A_{n-1}\), and let \(L\) be its side. Now, let \(P\) be a point anywhere on the inscribed circumference of \(A\).

If we know that \(\overline{A_0P}^2+\overline{A_1P}^2+\cdots+\overline{A_{n-1}P}^2=(45+24\sqrt{3})L^2\), find \(n\), i.e., the number of sides of \(A\).

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