CN-TST 2022 Problem 21

Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$ , the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$ .

Note: The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$ .