
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $a_1, a_2,...,a_{51}$  be non-zero elements of a field of characteristic  $p$ . We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence  $b_1, ... , b_{51}$ . If this new sequence is a permutation of the original one, find all possible values of  $p$ .

Present the answer in LaTex format: \boxed{Your answer}
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