On a characterization of abelian groups $G$ based on special commutator relations ($\exists n\in\Bbb N$ s.t. $[x^n,y]=[x,y^{n+1}],\forall x,y \in G$).

Question

Let $G$ be a group. If $\exists n\in \mathbb N$ such that $x^n=yx^n(y^{n+1}x)^{-1}xy^n,\forall x,y\in G$, then how to prove that $G$ is abelian?

Thoughts: the condition is same as saying $[x^n,y]=[x,y^{n+1}],\forall x,y \in G$, where $[a,b]:=a^{-1}b^{-1}ab$.