Cards. Why Does It Have To Be Cards?

On each of \(2013\) cards is written a unique number, i.e. all of the numbers are different. The cards are flipped face down (the numbers cannot be seen). In one go Andrei can choose ten cards, and Kostya (who knows what the numbers on the cards are), will tell him one of the numbers (Andrei will not know which one).

What is the largest \(t\) for which Andrei can guarantee to find \(t\) cards and know which number is written on the back of each one?

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