HMMT 2019 U7

For any real number $\alpha$, define

$$\operatorname{sign}(\alpha)= \begin{cases}+1 & \text { if } \alpha>0 \\ 0 & \text { if } \alpha=0 \\ -1 & \text { if } \alpha<0\end{cases}$$

How many triples $(x, y, z) \in \mathbb{R}^{3}$ satisfy the following system of equations

$$\begin{aligned} & x=2018-2019 \cdot \operatorname{sign}(y+z) \\ & y=2018-2019 \cdot \operatorname{sign}(z+x) \\ & z=2018-2019 \cdot \operatorname{sign}(x+y) ? \end{aligned}$$