Senior Kangaroo 2018 Q6

The line segments $PQRS$ and $WXYS$ intersect circle $C_1$ at points $P$, $Q$, $W$ and $X$.

The line segments intersect circle $C_2$ at points $Q$, $R$, $X$ and $Y$. The lengths $QR$, $RS$ and $XY$ are 7, 9 and 18 respectively. The length $WX$ is six times the length $YS$.

What is the sum of the lengths of $PS$ and $WS$?