KöMaL 2016 A663

There are given two positive integers: $k$ and $\ell$. A square with horizontal and vertical sides is divided into finitely many rectangles by line segments such that the following statements are satisfied: $(i)$ every horizontal or vertical line of the plane contains at most one of the segments; $(ii)$ no two segments cross each other in their interiors; $(iii)$ every horizontal line, intersecting the square but not containing any of the segments, intersects exactly $k$ rectangles; $(iv)$ every vertical line, intersecting the square but not containing any of the segments, intersects exactly $\ell$ rectangles. What can be the number of rectangles?