2019 Fermat Problem 22

In the diagram, two circles touch at P. Also, QP and SU are perpendicular diameters of the larger circle that intersect at O. Point V is on QP and VP is a diameter of the smaller circle. The smaller circle intersects SU at T, as shown. If QV = 9 and ST = 5, what is the sum of the lengths of the diameters of the two circles?

\textbf{(A)}\ 50\quad \textbf{(B)}\ 91\quad \textbf{(C)}\ 41\quad \textbf{(D)}\ 82\quad \textbf{(E)}\ 100

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.