Holiday Contest 2025 - Guts Round - Set 4 Problem 3

A rectangle with perimeter 44 is cut out from a square sheet of paper with side length a, such that the sides of the rectangle are parallel to the sides of the square and a vertex of the rectangle lies on top of a vertex of the square. This leaves a concave hexagon such that all interior angles are equal to 90 or 270 degrees, and the side length of the square is equal to the area of the hexagon. If the largest value of a can be expressed as \frac{\sqrt{p} + q}{r}, where p, q, r are all positive integers, and q is not divisible by the square of any prime, compute p + q + r.