TUYMAADA 2019 J8

Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$ , or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.