Factoring $2{^{3^{k+1}}}+1$

Question

I'm trying to factor $2{^{3^{k+1}}}+1=(X+1)(Y-2^{3^{k}}+1)$ and I don't know how to split the $2{^{3^{k+1}}}$ term into $X$ and $Y$. What is the trick here so that I can recognize this type of thing with other problems? This is for an induction problem I'm working on. The problem is to show that $n|(2^n+1)$ where $n$ is a power of $3$. Any hints or different approaches are welcome.