A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(5,1)$ and $(10,8)$. A line is drawn through points $A(0,0)$ and $B(4,2)$. Another line is drawn through points $C(0,9)$ and $D(4,7)$. How many distinct points on the perimeter of the rectangle lie on at least one of the two lines?