SMO 2019 Problem 4

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