Holiday Contest 2025 - Individual Round - Problem 3

A rectangular prism ABCDEFGH (vertices in that order) has a rectangular base ABCD which is parallel to base EFGH, such that BG, CF, DE, and AH are all perpendicular to these two bases. A right circular cone is constructed such that AB is the diameter of its circular base, the base is tangent to side GH, and the midpoint of CD coincides with the tip. It is known that AB = 12 and EH = 8. Given that the surface area of the entire solid formed by the union of the cone and the rectangular prism can be written as a+b\pi, for positive integers a and b, find the value of a+b.