Let $X$ and $Y$ be compact metric spaces. If $X^{\mathbb{N}}$ is homeomorphic to $Y^{\mathbb{N}}$ is $X$ homemorphic to $Y^k$?

Question

Let $X$ and $Y$ be compact metric spaces. If $X^{\mathbb{N}}$ is homeomorphic to $Y^{\mathbb{N}}$ (product topology) is $X$ homemorphic to $Y^k$ for $k \in \mathbb{N}$ or $k = \mathbb{N}$?

We must allow the possibility of $k = \mathbb{N}$ since the countable product of a Cantor set is a Cantor set.