
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  $p\geq 3$  be a prime number and  $0\leq r\leq p-3.$  Let  $x_1,x_2,\ldots,x_{p-1+r}$  be integers satisfying \[\sum_{i=1}^{p-1+r}x_i^k\equiv r \bmod{p}\]for all  $1\leq k\leq p-2.$  What are the possible remainders of numbers  $x_2,x_2,\ldots,x_{p-1+r}$  modulo  $p?$ *Proposed by Dávid Matolcsi, Budapest*

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