Is The Following Language Regular? Let $L_{1}$ and $L_{2}$ be 2 languages over the same alphabet $\Sigma$.  

$$A(L_1,L_2)=\{x\in \Sigma^*|\exists y,z\in L_2\text{ such that } yxz\in L_1\}$$

Assume that $L_{1}$ is regular and $L_{2}$ is context-free. The language $A(L_{1},L_{2})$:


is always a regular language
is always not a regular language
can sometimes be a regular language
cannot be context free


They say that the correct answer is 1.
Hint: Take a DFA for $L_1$. Check which states are reachable from the initial state via a word in $L_2$. Check from which states a final state can be reached via a word in $L_2$.