CN-TST 2019 Problem 16

Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ , such that

1. $f(0,x)$ is non-decreasing ;
2. for any $x,y \in \mathbb{R}$ , $f(x,y)=f(y,x)$ ;
3. for any $x,y,z \in \mathbb{R}$ , $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4. for any $x,y,a \in \mathbb{R}$ , $f(x+a,y+a)=f(x,y)+a$ .