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Let \(P = A_1, A_2 \dots A_k\) be a convex polygon on the plane. The vertices \(P = A_1, A_2 \dots A_k\) have integral coordinates and lie on a circle. Let \(S\) be the area of \(P\). An odd positive integer \(n\) is given such that the squares of the side lengths of \(P\) are integers divisible by \(n\). Prove that \(2S\) is an integer divisible by \(n\).