Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards.

Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens.

What is the largest number of chickens Pål can keep track of using this system?

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