import itertools

class Board:
    def __init__(self, width, height, quotient_x, quotient_y):
        self.width = width
        self.height = height
        self.quotient_x = quotient_x
        self.quotient_y = quotient_y
        self.lives = set()

    def toggle(self, x, y):
        if (x, y) in self.lives:
            self.lives.remove((x, y))
        else:
            self.lives.add((x, y))

    def evaluate(self):
        new_lives = set()
        for (x, y) in itertools.product(range(self.width), range(self.height)):
            neighbor_count = 0
            for (dx, dy) in itertools.product([-1, 0, 1], [-1, 0, 1]):
                if dx == 0 and dy == 0:
                    continue
                nx = x + dx
                ny = y + dy
                if self.quotient_x:
                    nx %= self.width
                if self.quotient_y:
                    ny %= self.height
                if (nx, ny) in self.lives:
                    neighbor_count += 1
            if (x, y) in self.lives and neighbor_count in [2, 3]:
                new_lives.add((x, y))
            if (x, y) not in self.lives and neighbor_count == 3:
                new_lives.add((x, y))
        self.lives = new_lives