Krdytkyu

I want to show that the language $L = $ {$a^mba^nba^p:m=n $ or $n = p$ or $m = p$} is either context-free or not context free by expressing the language as a union of three other languages $L_1$, $L_2$, and $L_3$.

By knowing if these three other languages are context-free/not context-free, I hope to indicate whether the language $L$ is context-free/not context-free.

Can anyone help me pick these three other languages that when unioned together form $L$?

Let $L_1={a^mba^nba^p:m=n}$. Consider the following grammar that generates this language with starting symbol $S$: $$
Slongrightarrow Cbtext{ }|text{ }Sa \
C longrightarrow btext{ }|text{ }aCa
$$
This grammar is context-free, so $L_1$ is context-free. A similar grammar will show that $L_2={a^mba^nba^p:n=p}$ is context-free.

Let $L_3={a^mba^nba^p:m=p}$. Consider the following grammar that generates this language with starting symbol $S$:
$$
Slongrightarrow bCbtext{ }|text{ }aSa \
C longrightarrow epsilontext{ }|text{ }aC
$$
This grammar is context-free, so $L_3$ is context-free. Since $L=L_1cup L_2cup L_3$, then $L$ is context-free.