MMO2025 Round 4

Let $K$ and $L$ be points on side $BC$, $M$ a point on side $AC$, and $N$ a point on side $AB$ of triangle $ABC$. These points are chosen such that $△KMC\sim △ABC$ and $△LBN\sim △ABC$. Suppose that the segments $NL$ and $KM$ intersect at point $P$ inside triangle $ABC$. The circumcircle of triangle $AMP$ intersects the line $CP$ again at point $X$, and the circumcircle of triangle $ANP$ intersects the line $BP$ again at point $Y$. Prove that the points $B,C,X$, and $Y$ lie on the same circle.