For any two languages \(L_1\) and \(L_2\) such that \(L_1\) is context-free…

For any two languages $L_1$ and $L_2$ such that $L_1$ is context-free and $L_2$ is recursively enumerable but not recursive, which of the following is/are necessarily true?

I. $\overline L_1$ (complement of $L_1$) is recursive

II. $\overline L_2$ (complement of $L_2$) is recursive

III. $\overline L_1$ is context-free

IV. $​​\overline L_1 \cup L_2$ is recursively enumerable