OLYMPIC-REVENGE 2021 Problem 2

Evan is a $n$ -dimensional being that lives in a house formed by the points of $\mathbb{Z}_{\geq 0}^n$ . His room is the set of points in which coordinates are all less than or equal to $2021$ . Evan's room has been infested with bees, so he decides to flush them out through $\textit{captures}$ . A $\textit{capture}$ can be performed by eliminating a bee from point $(a_1, a_2, \ldots, a_n)$ and replacing it with $n$ bees, one in each of the points: $$(a_1 + 1, a_2 , \ldots, a_n), (a_1, a_2 + 1, \ldots, a_n), \ldots, (a_1, a_2, \ldots, a_n + 1)$$ However, two bees can never occupy the same point in the house. Determine, for every $n$ , the greatest value $A (n)$ of bees such that, for some initial arrangement of these bees in Evan's room, he is able to accomplish his goal with a finite amount of $\textit{captures}$ .