Context-free grammar for $L = \{a^{2^k}, k \in\mathbb{N}\}$

Question

In an exercise, I am asked to find a context free grammar for language $L = \{a^{2^k}, k \in \mathbb{N}\}$.

I have been trying to use a "doubling" variable. If $a^{2n} \in L, n\in\mathbb{N}$ then use this variable to double the $a$'s that have been produced by the other language rules.

Is this thinking valid? So far I haven't been able to get anywhere with it, but it seems logical given the double-stack of powers.