Polygon Discovery

You are a member of the research team CPCI (Convex Polygon Computational Investigation), tasked with uncovering the properties of an unknown simple polygon $P$ located on a two-dimensional coordinate plane. The exact coordinates of the polygon $P$ are unknown, but you can use a special equipment that allows you to query the number of intersections between a given query line and $P$.

The polygon $P$ is known to be a convex polygon, meaning every line segment between two points inside the polygon is fully contained within its interior. Additionally, $P$ is non-degenerate, ensuring that its area is strictly greater than zero. The polygon also contains the origin $(0, 0)$ strictly within its interior. All vertices of $P$ have integer coordinates, with each coordinate lying within the range $[-1\,024, 1\,023]$ inclusive.

Write a program to compute the area of the polygon $P$ by invoking a series of queries.