Mis$`$ere one$-$pile Nim Consider the so$-$called mis$`$ere version of the one$-$pile Nim, in which the player taking the last chip loses the game. All the other conditions of the game remain the same, i$.$e$.,$ the pile contains n chips and on each move a player takes at least one but no more than m chips. Identify the winning and losing positions $($for the player to move next$)$ in this game.