Design PDA or CFG for the following language $L = \{wtw' \mid w,w' \in \{a, b\}^*, \hspace{0.3cm} |w|_a = |w'|_a, \hspace{0.3cm} |w|_b=2|w'|_b, \hspace{0.3cm} t\in\{c\}^* \}$.


  Note : $|w|_a$ means the number of $a$ in substring $w$.


To substring $w$, the number of symbol $a$ and $b$ is not relevant to each other.

It's also applied to substring $w'$.

To make matters worse, order of $a$ and $b$ (in both $w$ and $w'$) is not defined. So order can be anyway... It makes the problem harder for me.

I designed several PDAs but they resulted in contradiction.

So I tried to construct a CFG. But designing a CFG is more difficult for me.

Could you give me some hints to design PDA or CFG for the above language?
There is no such grammar: the language is not context-free.

$L\cap a^*b^*ca^*b^* = \{a^nb^{2m}ca^nb^m\mid m,n\ge 1\}$.