Holiday Contest 2025 - Team Round - Problem 20

Consider a circle \omega with center O. Let AB be a chord of \omega, not passing through O. Let the tangents to \omega at A, B intersect at P. Let C be on the minor arc AB with AC > BC > 0, and let PC intersect \omega at D \neq C. Define Q_1,Q_2 to be the intersections of (COD) and line segment AB (with Q_1 = Q_2 if there is only 1 intersection), and R = CD \cap AB. If PC = 12, PD = 20, and Q_1R = Q_2R = 5, find the sum of all possible values of r^2.