KöMaL 2025 B5469

There are three cities in Bergengócia, $A$, $B$, and $C$, inhabited by a total of 2025 people. Starting from the current year, at the end of each year, half of the inhabitants of $A$ move to $B$, a third of the inhabitants of $B$ move to $C$, and a quarter of the inhabitants of $C$ move to $A$ (if any of the values of these fractions is not an integer, we round the value up). Prove that after a few years, the number of inhabitants of all three cities will remain unchanged.