I am pretty desperate finding the correct context free grammar for the following language: $$L=\{a^lb^mc^n \mid l,m,n \in \mathbb{N}_0, \, l\geq 2n+m\}$$
I would really appreciate if anyone could tell me how to generate the CFG from this language as I am stuck on this one for 2 days now.
I can't find a solution for adding the right amount of $b$'s and $c$'s to a generated string.
At the moment I have: $$ \begin{align*} S &\rightarrow aS \mid T \mid U \mid \varepsilon \\ T &\rightarrow aTb \mid \varepsilon \\ U &\rightarrow aaUc \mid \varepsilon \end{align*}$$