In the cartesian plane, consider the subset of all the points with both integer coordinates. Prove that it is possible to choose a finite non-empty subset $S$ of these points in such a way that any line $l$ that forms an angle of $90^{\circ},0^{\circ},135^{\circ}$ or $45^{\circ}$ with the positive horizontal semi-axis intersects $S$ at exactly $2024$ points or at no points.