
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  $p \equiv 2 \pmod 3$  be a prime,  $k$  a positive integer and  $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$ . For any integer  $n$ , let  $R(n)$  denote the remainder when  $n$  is divided by  $p$  and let  $S = \{0,1,\cdots,p-1\}$ . At each step, you can either (a) replaced every element  $i$  of  $S$  with  $R(P(i))$  or (b) replaced every element  $i$  of  $S$  with  $R(i^k)$ . Determine all  $k$  such that there exists a finite sequence of steps that reduces  $S$  to  $\{0\}$ .

*Proposed by fattypiggy123*

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