Creating a CFG from a specific CFL 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:
$$ \begin{align*}
S &amp;\rightarrow aS \mid T \mid U \mid \varepsilon \\
T &amp;\rightarrow aTb \mid \varepsilon \\
U &amp;\rightarrow aaUc \mid \varepsilon
\end{align*}$$
It might be easier than you have been thinking.

$$\begin{aligned} 
S&amp;\rightarrow aS\mid U\\
U&amp;\rightarrow aaUc\mid T\\ 
T&amp;\rightarrow aTb\mid \epsilon 
\end{aligned}$$

The idea is to identify the language that could be obtained by a fixed operation that peels off some symbols. Peeling off $a$'s, we shrink words in $S$ to words in $U$. Peeling off some $a$'s and $c$'s, we shrink words in $U$ to words in $T$.
$$\begin{aligned} 
S&amp;=\{a^ka^{2n}a^mb^mc^n \mid k,m,n \in \mathbb{N}_0\}\\ 
U&amp;=\{a^{2n}a^mb^mc^n \mid m,n \in \mathbb{N}_0\}\\ 
T&amp;=\{a^mb^m\mid m \in \mathbb{N}_0\}
\end{aligned}$$