KöMaL 2018 B4992

To start, each of the numbers $1,2,\ldots, n$ is coloured either red or blue. In every move, three distinct numbers in arithmetic progression are selected, and the colour of each of the three numbers is changed. For what values of $n$ is it true that however the numbers $1,2,\ldots, n$ are coloured initially, it is possible to turn all of them red by applying an appropriate sequence of such steps?