Question 144
Suppose $PQRS$ is a convex quadrilateral with $\angle SPQ = \angle PQR = 120^\circ$, $SP - QR = 36$, $RS = 84$, and $QR$ is a positive even integer. Let $T$ be the intersection of lines $SP$ and $QR$. What is the largest possible perimeter of $\triangle PQT$?