Let $F(N)$ be the maximum number of lattice points in an axis-aligned $N\times N$ square that the graph of a single strictly convex increasing function can pass through.

You are given that $F(1)=2$, $F(3)=3$, $F(9)=6$, $F(11)=7$, $F(100)=30$ and $F(50000)=1898$.
Below is the graph of a function reaching the maximum $3$ for $N=3$:

Find $F({10}^{18})$.