Let $n$ be a positive integer. On a $2n\times 2n$ board, the $2n^2$ squares were painted white and the other $2n^2$ squares were painted black. One operation is to choose a $2\times 2$ subtable and mirror its $4$ cells about the vertical or horizontal axis of symmetry of that subtable. For what values of $n$ is it always possible to obtain a chess-like coloring from any initial coloring?