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 $90^\circ$ or $270^\circ$, 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 $\frac{p\sqrt{q} + r}{s}$, where $p, q, r,s$ are all positive integers, $q$ is not divisible by the square of any prime, and $\gcd(p,r,s)=1$, compute $p + q + r+s$.