Proving a language is not regular using Myhill Nerode Theorem Let $L = \{\alpha\in\{a,b,c\}^{*} \mid \alpha \text{ is palindrome}\}$, show that $L$ is not regular using Myhill-Nerode relation.

I don't know how to show that $L$ has infinite equivalence classes because $\alpha$ is a palindrome. I tried to use something like this, but I don't know if its correct:

$\alpha \equiv_{L} \beta \iff \alpha (aba)^k \in{L} \iff \beta (aba)^k \in{L}$ $\forall k \in \mathbb N$ which implies that for every k there exists an equivalence class because the repetition of aba k times.
Take $a^nb$ and $a^mb$ for n != m. Since the first can be followed by $a^n$ and the second can’t, they are in different equivalence classes. Therefore the number of equivalence classes is not finite. 

C isn’t even needed.