A board that has some of its squares painted black is called acceptable if there are no four black squares that form a $2 \times 2$ subboard. Find the largest real number $\lambda$ such that for every positive integer $n$ the following proposition holds: mercy: if an $n \times n$ board is acceptable and has fewer than $\lambda n^2$ dark squares, then an additional square black can be painted so that the board is still acceptable.