CH-SMO 2022 Problem 4

Let $n \geq 2$ be an integer. Switzerland and Liechtenstein are performing their annual festive show. There is a field divided into $n \times n$ squares, in which the bottom-left square contains a red house with $k$ Swiss gymnasts, and the top-right square contains a blue house with $k$ Liechtensteiner gymnasts. Every other square only has enough space for a single gymnast at a time. Each second either a Swiss gymnast or a Liechtensteiner gymnast moves. The Swiss gymnasts move to either the square immediately above or to the right and the Liechtensteiner gymnasts move either to the square immediately below or to the left. The goal is to move all the Swiss gymnasts to the blue house and all the Liechtensteiner gymnasts to the red house, with the caveat that a gymnast cannot enter a house until all the gymnasts of the other nationality have left. Determine the largest $k$ in terms of $n$ for which this is possible.