Trapezoid Shuffle

Level pending

The sum of the areas of all possible trapezia (quadrilaterals with one pair of parallel sides) with sides of length 3, 3, 5, and 7 (not necessarily in that order) can be written as \(a+b\sqrt{c}\), where a, b, and c are integers, and c is positive and square-free. Find \(a+b+c\).

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