Nemotron-IMO-Bench 2026 G13

The $100\times 100$ square unit lattice $\textbf{L}$ has $10000$ points. A set $\mathcal{F}$ of polygons has the property that the set of vertices of the union of all polygons in $\mathcal{F}$ is exactly $\textbf{L}$. Find the maximum possible sum of the areas of the polygons in $\mathcal{F}$.