If there exists a positive integer $n$ such that any element $x$ of a ring $R$ satisfies $x^{4^n+2} = x$, then every element $x$ in $R$ is idempotent

Let $R = (A,+,\cdot)$ be a ring. If there exists a positive integer $n$ such that any element $x$ of a ring $R$ satisfies $x^{4^n+2} = x$, then every element $x$ in $R$ is idempotent.

I have studied some group theory but I don't think it's that helpful here. Basically by multiplying both sides a bunch of times with $x^{4^n + 1}$ you can show that $$x^{k(4^n + 1) + 1} = x \hspace{5px} \forall k \in \mathbb{N}.$$

My intuition goes along the lines of "if $x^a = x$ and $x^b = x$ then $x^{(a,b)} = x$" but in the absence of multiplicative inverses that wouldn't work (the reason I thought this might have been helpful is because the exponent of $2$ in $4^n + 2$ is always $1$, so by finding an appropriate $k$ we could make the gcd $2$).

How should I proceed?

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