TSTST 2026 Problem 6

Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $S$ be a point inside triangle $ABC$. Let $P \neq S$ be the second intersection of line $AS$ with the circumcircle of $BCS$.

Let $T \neq A$ be a variable point on $\Gamma$, and let $Q \neq A$ and $R \neq A$ be the second intersections of line $AT$ with the circumcircles of $ABP$ and $ACP$, respectively. Let $\omega$ be the circumcircle of $PQR$. Show that, as $T$ varies along $\Gamma$, the intersection(s) of line $ST$ with $\omega$ lie on two fixed circles (excluding cases where $\omega$ is undefined).