Let $C$ be the set of points $(x,y)$ with integer coordinates in the plane where $1\leq x\leq 900$ and $1\leq y\leq 1000$. A polygon $P$ with vertices in $C$ is called emerald if $P$ has exactly zero or two vertices in each row and each column and all the internal angles of $P$ are $90^\circ$ or $270^\circ$. Find the greatest value of $k$ such that we can color $k$ points in $C$ such that any subset of these $k$ points is not the set of vertices of an emerald polygon. On the left, an example of an emerald polygon; on the right, an example of a non-emerald polygon.