Let $L_1 \subseteq \{0,1\}^{*}$ be a regular language, and let $L_2 \subseteq \{0,1\}^{*}$ be some (not necessarily regular) language. Show that $$L=\left\{ \sigma_{1}\#\sigma_{2}\dots\#\sigma_{n}\mid\substack{n\geq1\\ \sigma_{i}\in\left\{ 0,1\right\} \\ \exists w_{1},\dots w_{n-1}\in L_{2}\quad\sigma_{1}w_{1}\sigma_{2}w_{2}\dots w_{n-1}\sigma_{n}\in L_{1} } \right\} $$ is regular.
I tried thinking about the Myhill–Nerode classes, and about closure under homomorphism, and homomorphism inverse, but I didn't achieve any result. I also tried thinking about $L_1$'s DFA and trying to squeeze NFA in between edges, but I don't have an NFA for L2...