Unknown polygon!

Geometry Level 4

Let \[ A_1 , A_2,A_3, \ldots, A_n \] be the vertices of a regular \(n\)-sided polygon inscribed in a circle of radius \(R\). If \[(A_1 A_2)^2 + (A_1 A_3)^2 +\cdots + ( A_1 A_n) ^2 = 14R^2,\] find the number of sides of the polygon.

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