2013 AIME I Problem 12

2013 AIME I 12. Let \Delta PQR be a triangle with \angle P = 75^\circ and \angle Q = 60^\circ. A regular hexagon ABCDEF with side length 1 is drawn inside \Delta PQR so that side \overline{AB} lies on \overline{PQ}, side \overline{CD} lies on \overline{QR}, and one of the remaining vertices lies on \overline{RP}. There are positive integers a, b, c, and d such that the area of \Delta PQR can be expressed in the form \frac{a + b\sqrt{c}}{d}, where a and d are relatively prime and c is not divisible by the square of any prime. Find a + b + c + d.

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