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I'm working on a proof that deals with a modified PDA, which is identitical to a PDA, but with transitions: \begin{equation*} a,b \to c \end{equation*} where $a$ is a context free grammar, instead of just a symbol. The modified PDA then reads a prefix $w'$ of $w$, the suffix of the input stream, where $w' \in L(a)$.

I'm try to prove that the language recognized by this modified PDA is context free, but I'm really stuck. I have no idea how to even get started. If someone could point me in the right direction, I would really appreciate it.

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Hint: Try replacing each such transition by an appropriate PDA.

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