HMMT 2019 U31

Let $A B C$ be a triangle with $A B=6, A C=7, B C=8$. Let $I$ be the incenter of $A B C$. Points $Z$ and $Y$ lie on the interior of segments $A B$ and $A C$ respectively such that $Y Z$ is tangent to the incircle. Given point $P$ such that

$$\angle Z P C=\angle Y P B=90^{\circ}$$

find the length of $I P$.