There's a board with $$$n \times m$$$ white squares. Jeroen plays a single-player game on this grid. Each turn he colors one square black. Two squares are adjacent if they share an edge. The squares, together with these adjacency relations form a planar graph. The game ends when a simple cycle of black squares is formed that has a non-empty interior. By drawing the edges of the graph with straight lines, between the centres of the squares we get a drawing of the planar graph. A square that does not lie on the cycle, is in the interior of the cycle, if it lies in the interior of the polygon that the edges of the cycle form, when drawn as straight line segments. Jeroen wants to produce a pretty picture on the playing board when he's done so he already thought of the moves he is going to play. If a move happens to end the game, he doesn't play it. In the input there are $$$k$$$ distinct positions $$$(r_i,c_i)$$$, the moves that Jeroen has planned out.

Your job is to figure out for each move, if it would end the game or not, and print a bitstring with $$$1$$$ if the $$$i$$$-th move ends up being played, and $$$0$$$ if this move would end the game, and does not get played.