from itertools import product, chain

class DynamicProduct:

    def __init__(self, n):
        self.pools = [[] for _ in range(n)]
        self.gen = (x for x in [])
        self.exclusions = []

    def next(self):
        while True:
            res = next(self.gen, None)
            if res is None:
                return None
            bad = any(predicate(res) for predicate in self.exclusions)
            if not bad:
                return res

    def add(self, index, elem):
        self.pools[index].append(elem)
        gens = []
        for i in range(len(self.pools)):
            if i != index:
                gen = (x for x in self.pools[i])
            else:
                gen = [elem]
            gens.append(gen)
        prod = product(*gens)
        self.gen = chain(self.gen, prod)

    def delete(self, index, elem):
        self.pools[index].remove(elem)
        self.exclude(lambda x: x[index] == elem)

    def exclude(self, predicate):
        self.exclusions.append(predicate)