For each positive integer $k$, find all positive integer $n$ such that there exists a permutation $a_1,\ldots,a_n$ of $1,2,\ldots,n$ satisfying $$a_1a_2\ldots a_i\equiv i^k \pmod n$$for each $1\le i\le n$. (Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)