OPM 2024 Problem 4

A circle inscribed in the square $ABCD$ , with side $10$ cm, intersects sides $BC$ and $AD$ at points $M$ and $N$ respectively. The point $I$ is the intersection of $AM$ with the circle different from $M$ , and $P$ is the orthogonal projection of $I$ into $MN$ . Find the value of segment $PI$ .