Where Do They Intersect?

Geometry Level 3

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.

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