Holiday Contest 2025 - Individual Round - Problem 8

Regular hexagon ABCDEF with side length 1 is inscribed in a rectangle WXYZ, such that A lies on size WX, C lies on side XY, D lies on side YZ, and F lies on side WZ. As well, E and B lie in the interior of WXYZ. Given that the area of rectangle WXYZ is \sqrt 3 + \frac{48}{25}, the sum of all possible values of the area of triangle ABX can be written as \frac{a-b\sqrt c}{d}, where a, b, c, and d are positive integers such that c is not divisible by the square of any prime and \gcd(a,b,d)=1. Find a+b+c+d.