On a table, there are $100$ red and $k$ white buckets, all initially empty. In each move, a red and a white bucket are selected, and an equal amount of water is added to both. After a certain number of moves, no bucket is empty, and for every pair of buckets that were selected together at least once, the amount of water in these buckets is the same. Find all possible values of $k$.

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