2019 Gauss 8 Problem 25

In quadrilateral PQRS, diagonals PR and SQ intersect at O inside PQRS, SP = SQ = SR = 1, and \angle QSR = 2\angle QSP. Marc determines the measure of the twelve angles that are the interior angles of \triangle POS, \triangle POQ, \triangle ROS, and \triangle ROQ. The measure of each of these angles, in degrees, is a positive integer, and exactly six of these integers are prime numbers. How many different quadrilaterals have these properties and are not rotations or translations of each other?

\textbf{(A)}\ 7\quad \textbf{(B)}\ 5\quad \textbf{(C)}\ 9\quad \textbf{(D)}\ 6\quad \textbf{(E)}\ 8

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 Gauss contests here.