There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20\%$ of cat-owners also own a dog, $70\%$ of dog-owners do not own a cat, and $50\%$ of people who don’t own a cat also don’t own a dog, how many people own neither type of pet?

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