Alice has two bags. each bag has $4$ slips of paper with the numbers $1$ through $4$ on them. betty also has two bags, each with $4$ slips of paper with positive integers on them. they decide to play a game whereby each girl pulls a slip from each of her own bags, records the sum of the numbers, then returns each slip to the bag it came from. the numbers in betty's bags are not $1$ through $4$ in each bag, but the expected distribution of her sums is the same as alice's. what are all eight numbers in betty's bags?