FKMO 2019 Problem 6

A sequence $\{x_n \}=x_0, x_1, x_2, \cdots$ satisfies $x_0=a(1\le a \le 2019, a \in \mathbb{R})$ , and $$x_{n+1}=\begin{cases}1+1009x_n &\ (x_n \le 2) 2021-x_n &\ (2<x_n \le 1010) 3031-2x_n &\ (1010<x\le 1011) 2020-x_n &\ (1011<x_n) \end{cases}$$ for each non-negative integer $n$ . If there exist some integer $k>1$ such that $x_k=a$ , call such minimum $k$ a fundamental period of $\{x_n \}$ . Find all integers which can be a fundamental period of some seqeunce; and for such minimal odd period $k(>1)$ , find all values of $x_0=a$ such that the fundamental period of $\{x_n \}$ equals $k$ .