Atiab has more ducks than he can manage to keep track of. Therefore, he keeps an index card for each duck. He keeps the cards in $12$ boxes, each of which has room for $3484$ cards.

Unfortunately, Atiab has a forgetful mind, 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 $7$ boxes, no matter which $7$, these $7$ boxes taken together will contain at least one card for each of his ducks.

What is the largest number of ducks Atiab can keep track of using this system?