PL-OM 2024 Problem 2

Let $n$ be a positive integer. Bolek draws $2n$ points in the plane, no two of them defining a vertical or a horizontal line. Then Lolek draws for each of these $2n$ points two rays emanating from them, one of them vertically and the other one horizontally.

Lolek wants to maximize the number of regions in which these rays divide the plane. Determine the largest number $k$ such that Lolek can obtain at least $k$ regions independent of the points chosen by Bolek.