Creating a CFG from a specific CFL
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 $T$.
$$\begin{aligned} 
S&amp;=\{a^ka^{2n}a^mb^mc^n \mid k,m,n \in \mathbb{N}_0\}\\ 
U&amp;=\{a^{2n}a^mb^mc^n \mid m,n \in \mathbb{N}_0\}\\ 
T&amp;=\{a^mb^m\mid m \in \mathbb{N}_0\}
\end{aligned}$$