Is there a model of this equational theory which is not power-associative?

Question

This is a follow up to my previous question, here: Two questions regarding equational axiomatizations of power-associative magmas.. As before, let $t$ be a term, in the sense of universal algebra. I define the function $L(t,m)$, where $m$ is a positive integer, recursively by $L(t,1)=t$ and $ L(t,m+1)=L(t,m)* t$. Dually, I define the function $R(t,m)$ recursively by $R(t,1)=t$ and $R(t,m+1)=t*R(t,m)$. $L$ and $R$ stand for left and right, respectively. Now, my question. Is there a magma $(M;*)$ which satisfies the equations $L(x,m)=R(x,m)$, for $m$ ranging over all positive integers, but which is not power-associative? If so, can someone exhibit such a magma?

Answer 1

Let $(M;*)=(\{0,1\};*)$ where $x*y:=x\;\textrm{NOR}\;y$.
($x\;\textrm{NOR}\;y = \neg(x\vee y)$.)

The NOR operation is commutative, so all $L=R$ equations will hold. But $(0*0)*(0*0)\neq 0*(0*(0*0))$.