Triangle-Ception II

Geometry Level 5

Quadrilateral \(ABCD\) is an isosceles trapezoid with \(AB=6,BC=6,CD=12,AD=6\).

Points \(X\) and \(Y\) are chosen on sides \(AD\) and \(DC\) respectively, such that \(DX=q\) and \(DY=p\).

If the area of triangle \(XYB\) is \(11\sqrt{3}\), the minimum value of \(p-q\) can be expressed in the form \(m\sqrt{n}\), where \(n\) is not divisible by the square of any prime. Find \(m+n\).

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