Where Do They Intersect?

Let $\Gamma_1$ and $\Gamma_2$ be two concentric circles, and let the radius of $\Gamma_1$ be greater than that of $\Gamma_2.$ Consider two non-parallel lines $\ell_1$ and $\ell_2$ passing through $\Gamma_2.$ One of the intersections of $\ell_1$ and $\Gamma_2$ is $P;$ one of the intersections of $\ell_2$ and $\Gamma_2$ is $Q.$ Let $R$ be one of the intersections of $\ell_1$ and $\Gamma_1,$ and let $S$ be the second intersection point of $\Gamma_1$ and the circumcircle of $\triangle PQR.$ Suppose $S$ lies on $\ell_2.$ Lines $\ell_1$ and $\ell_2$ meet at $X.$ Suppose $XP= XQ.$ Where does $X$ lie?

I did not include a picture because I'm afraid that even an inaccurate diagram might give the answer away.