
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 \( p \) be a prime number such that \( p \equiv 2 \mod 3 \), and \( k \) be a positive integer. Consider the polynomial
\[ 
P(x) = 3x^{\frac{2p-1}{3}} + x^{\frac{p+1}{3}} + x + 1.
\]
For any integer \( n \), let \( R(n) \) represent the remainder when \( n \) is divided by \( p \). Define the set \( S = \{0, 1, \cdots, p-1\} \). The elements in set \( S \) are replaced as follows: Replace each element \( i \) in \( S \) with \( R(P(i)) \) or replace each element \( i \) in \( S \) with \( R(i^k) \). Determine all \( k \) such that after a finite number of replacements, the set \( S \) becomes \( \{0\} \).

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