A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(1,1)$ and $(4,3)$. A line is drawn through points $E(0,0)$ and $F(5,2)$. Another line is drawn through points $G(0,5)$ and $H(3,4)$. How many distinct points on the perimeter of the rectangle lie on at least one of the two lines?