How to prove that every countable compact Hausdorff space is homeomorphic to a well-ordered set with its order topology?

Question

I encountered this problem in a textbook: Let $X$ be a countable compact Hausdorff topological space, I was asked to prove that $X$ is always homeomorphic to a (necessarily countable) topological space $Y$ equipped with the order topology of a certain well-order $\leqslant$ on $Y$ (a well-ordering on $Y$ is a total order $\leqslant$ on $Y$ such that every subset of $(Y,\leqslant)$ admits a minimal element, or equivalently, an ordinal). But I had no ideal where to start (I guessed that firstly I had to prove $X$ is first countable, but I failed). Can anyone help me about it? Any useful hint will be appreciated.