CMO 2017 Problem 4

Let $n \geq 2$ be a natural number. For any two permutations of $(1,2,\cdots,n)$ , say $\alpha = (a_1,a_2,\cdots,a_n)$ and $\beta = (b_1,b_2,\cdots,b_n),$ if there exists a natural number $k \leq n$ such that $$b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; a_i, & \text{} k < i \leq n, \end{cases}$$ we call $\alpha$ a friendly permutation of $\beta$ .

Prove that it is possible to enumerate all possible permutations of $(1,2,\cdots,n)$ as $P_1,P_2,\cdots,P_m$ such that for all $i = 1,2,\cdots,m$ , $P_{i+1}$ is a friendly permutation of $P_i$ where $m = n!$ and $P_{m+1} = P_1$ .