It's Geometry !!!

Geometry Level 4

\( ABCD \) is a square where \(AB=4.\) \(P\) is a point inside the square such that \(\angle PAB = \angle PBA = 15^\circ.\) \(E\) and \(F\) are the midpoints of \(AD\) and \(BC\) respectively\(.\) \(EF\) intersects \(PD\) and \(PC\) at points \(M\) and \(N\) respectively\(.\)

\(Q\) is a point inside the quadrilateral \(MNCD\) such that \(\angle MQN = 2\angle MPN.\) The perimeter of the \(\triangle MNQ\) is \(\frac{3\sqrt{5}+8\sqrt{3} -8}{2\sqrt{3}}. \) \(PQ^{2}\) can be written as \(\frac{a}{b}\) ; where \(a\) and \(b\) are co-prime positive integers\(.\) Find the value of \(a+b.\)

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