SHARYGIN 2021 Problem 98

A quadrilateral $ABCD$ is circumscribed around a circle $\omega$ centered at $I$ . Lines $AC$ and $BD$ meet at point $P$ , lines $AB$ and $CD$ meet at point $£$ , lines $AD$ and $BC$ meet at point $F$ . Point $K$ on the circumcircle of triangle $E1F$ is such that $\angle IKP = 90^o$ . The ray $PK$ meets $\omega$ at point $Q$ . Prove that the circumcircle of triangle $EQF$ touches $\omega$ .