OFM 2021 S3

Let $ABCD$ be a square with incircle $\Gamma$ . Let $M$ be the midpoint of the segment $[CD]$ . Let $P \neq B$ be a point on the segment $[AB]$ . Let $E \neq M$ be the point on $\Gamma$ such that $(DP)$ and $(EM)$ are parallel. The lines $(CP)$ and $(AD)$ meet each other at $F$ . Prove that the line $(EF)$ is tangent to $\Gamma$