It is really a Brilliant symbol

Geometry Level 5

A regular \(12\)-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the \(12\)-gon can be written in the form of \(p+q\sqrt{2}+r\sqrt{3}+s\sqrt{6}\), where \(p,q,r\) and \(s\) are positive integers. Find \(p+q+r+s\).

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