HMMT 2018 T10

Let $n$ and $m$ be positive integers which are at most $10^{10}$. Let $R$ be the rectangle with corners at $(0,0),(n, 0),(n, m),(0, m)$ in the coordinate plane. A simple non-self-intersecting quadrilateral with vertices at integer coordinates is called far-reaching if each of its vertices lie on or inside $R$, but each side of $R$ contains at least one vertex of the quadrilateral. Show that there is a far-reaching quadrilateral with area at most $10^{6}$. (A side of a rectangle includes the two endpoints.)