ELMO SL 2019 N5

Given an even positive integer $m$ , find all positive integers $n$ for which there exists a bijection $f:[n]\to [n]$ so that, for all $x,y\in [n]$ for which $n\mid mx-y$ , $$(n+1)\mid f(x)^m-f(y).$$ Note: For a positive integer $n$ , we let $[n] = \{1,2,\dots, n\}$ .

Proposed by Milan Haiman and Carl Schildkraut