Let ${n}$ be a fixed positive integer. ${A}$ and ${B}$ play the following game: $2023$ coins marked $1, 2, \dots, 2023$ lie on a circle (the marks are considered in module $2023$) and each coin has two sides. Initially, all coins are head up and ${A}$'s goal is to make as many coins with tail up. In each operation, ${A}$ choose two coins marked ${k}$ and $k+3$ with head up (if ${A}$ can't choose, the game ends) and ${B}$ choose a coin marked $k+1$ or $k+2$ and flip it. If at some moment there are ${n}$ coins with tail up, ${A}$ wins. Find the largest ${n}$ such that ${A}$ has a winning strategy.