There are $ 5$ dogs, $ 4$ cats, and $ 7$ bowls of milk at an animal gathering. Dogs and cats are distinguishable, but all bowls of milk are the same. In how many ways can every dog and cat be paired with either a member of the other species or a bowl of milk such that all the bowls of milk are taken?

Let's think step by step and output the final answer within \boxed{}.
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