$Ax=y$ with a inequality condition Prove every $x_i$ is positive

Question

Let $A=(a_{ij})\in M_n(\mathbb R)$.

Let $x =(x_1,\cdots,x_n)^t,\ y=(y_1,\cdots,y_n)^t$ be vectors satisfying $Ax=y$ and $$\sum_{j\ne i} \max\{a_{ij},0\}<y_i\le a_{ii}+ \sum_{j\ne i}\min\{a_{ij},0\},\forall i=1,\cdots,n$$ Prove that $\forall i,\ x_i>0$.