HMMT 2004 G10

Right triangle $X Y Z$ has right angle at $Y$ and $X Y=228, Y Z=2004$. Angle $Y$ is trisected, and the angle trisectors intersect $X Z$ at $P$ and $Q$ so that $X, P, Q, Z$ lie on $X Z$ in that order. Find the value of $(P Y+Y Z)(Q Y+X Y)$.