# 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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