HMMT 2020 T7

Positive real numbers $x$ and $y$ satisfy

$$||\cdots|||x|-y|-x| \cdots-y|-x|=||\cdots|||y|-x|-y| \cdots-x|-y|$$

where there are 2019 absolute value signs $|\cdot|$ on each side. Determine, with proof, all possible values of $\frac{x}{y}$.