Albert makes a sequence of numbers, the first 2023 of which are exactly the numbers 1 through 2023, in some order. To determine each subsequent number, Albert takes the median of the 2023 previous numbers. You find the median of 2023 numbers by sorting them by size and taking exactly the middle number from that sequence. How many different values can the three thousandth number of Albert's sequence have?
This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.