Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions: All squares touching the border of the grid are coloured black. No four squares forming a $2\times 2$ square are coloured in the same colour. No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching squares have the same colour. Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?