Using CFL pumping lemma to show that a language is not context free

Question

consider the language: $$L=\left\{ w*a^n:w\in\left\{0,1 \right\}^*\text{ is the binary representation of the number }n\right\} $$ over the alphabet $\Sigma=\left\{0,1,*,a \right\}$.
Is $L$ a context-free language? If it is show a CFG $G$ such that $L(G)=L$, or use the pumping lemma for CFL to disprove.
I have a very strong intuition that $L$ is not context free, but I'm a bit stuck on choosing the correct $w$ in the pumping lemma. I would be very grateful if someone could guide me on how to choose $w$.