In Tigre there are $2024$ islands, some of them connected by a two-way bridge. It is known that it is possible to go from any island to any other island using only the bridges (possibly several of them). In $k$ of the islands there is a flag ($0 \le k \le 2024$). Ana wants to destroy some of the bridges in such a way that after doing so, the following two conditions are met: \\ $\bullet$ If an island has a flag, it is connected to an odd number of islands. \\ $\bullet$ If an island does not have a flag, it is connected to an even number of islands. \\ Determine all values of $k$ for which Ana can always achieve her objective, no matter what the initial bridge configuration is and which islands have a flag.