Alexandra, Bob, Chelsea, and Don are all logic experts who always tell the truth.

They each ate some numbers of apples: $a, b, c,$ and $d,$ respectively. They don't know the numbers the others ate, but they do know that each person ate **at least 1 apple** and that they ate **11 apples in total**.

Alexandra says, “Bob, I don't know whether you ate more than I did. Did you?”

Bob replies, “I don’t know. Chelsea, I also don't know whether you ate more than I did. Did you?”

Chelsea replies, “I don’t know.”

Then Don says, "Oh, now I know how many each of us ate."

How many apples did each person eat? Give your answer as the concatenation of their numbers: $\overline{abcd}.$

**Assume** that the people all have perfect knowledge about each others' logical ability and honesty. That is, they all know that they are all logic experts that always tell the truth, and they all know that they know this, and so on.

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