Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide the sum of all possible characteristics.

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