How to prove this language is context free?
I see that the language is equivalent to $L = \{aa^nbbb^mccccc^n\}$,


Indeed


  but I'm not sure if that helps at all.


Well, maybe not. How about if we write the equivalent language ever-so-slightly differently:

$$L = \{a^nabbb^mccccc^n\}$$

That's the same as

$$\begin{align}L &amp;= \{a^nMc^n\}\\
M &amp;= \{abbb^mcccc\}\\
\end{align}$$
And then
$$\begin{align}L &amp;= \{a^nMc^n\}\\
M &amp;= \{abbNcccc\}\\
N &amp;= \{b^m\}\\
\end{align}$$
Which seems pretty straight-forward.

Or, to put it another way: Look for symmetries. Or try to create them.