Context-free grammar from language

Question

I'm trying to come up with a context-free grammar for the following language:

$$L = \{a^mb^nc^{m+n}\mid 0 \le n \le m\}$$

My thinking is that i can rewrite this to

$$L = \{a^mb^nc^nc^m\mid 0 \le n \le m\}$$

and then create the grammar

$$ \begin{align*} S &\rightarrow aSc \mid B\\ B &\rightarrow bBc \mid \epsilon \end{align*} $$

The problem is that this grammar doesn't capture $0 \le n \le m$ and I can't figure out how to do this. Any help would be greatly appreciated.

Answer 1

Hint: is the language $L' = \{a^n b^n c^{2n} \mid n \ge 0\}$ context-free?