Find all n for which $2^n$ divides $3^n-1$

Question

Find all positive integers such that $2^n$ divides $3^n-1$.

I tried factoring $3^n-1$ as such: $$3^n-1=(3-1)(3^{n-1}+3^{n-2}+3^{n-3}+\cdots+3+1).$$ But now I'm stuck.