Putnam 2022 B4

Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1,\dots,x_n$ such that each of the sets

$$\begin{gathered}\{x_1,x_2,x_3\}, \{x_2,x_3,x_4\}, \dots, \\ \{x_{n-2},x_{n-1},x_n\}, \{x_{n-1},x_n, x_1\}, \text{ and } \{x_n, x_1, x_2\}\end{gathered}$$

forms a 3-term arithmetic progression when arranged in increasing order.