If $4^n + 2^n + 1$ is prime, then $n$ must be a power of $3$

Question

I was messing around with some number theory when I stumbled upon this. The statement I'm trying to prove is,

If $4^n + 2^n + 1$ is prime, then $n$ must be a power of $3$

I think it is true but I'm not entirely sure how to go about proving it. Any hints to start off?

I tried reducing the problem to something of the form $2^{2n} + 2^{n} + 1$ and messing with mods but to no avail. I'm not actually convinced it's true myself which is a difficult position to prove something from.