Proving an idempotent binary operation where $(x\ast y)\ast z=(y\ast z)\ast x$ is commutative

Question

Let $S$ be a set and $\ast$ be a binary operation on $S$ satisfying

1) $x\ast x=x$ for all $x\in S$,

2) $(x\ast y)\ast z=(y\ast z)\ast x$ for all $x,y,z \in S$.

Show that $x\ast y=y\ast x$.

I haven't got any idea for solving this problem.

Answer 1

$$(xy)x=(yx)x=(xx)y=xy$$ $$xy=(xy)(xy)=(y(xy))x=((xy)x)y=(xy)y=(yy)x=yx$$