Advanced Number Theory · Challenge problem

A path from $(0,0)$ to $(10,10)$ consists of $20$ unit steps, each either to the right or up. Let $N$ be the number of such paths that never pass through a point $(x,y)$ with $y - x \ge 3$ (so the path never rises more than $2$ above the diagonal $y = x$). Find the remainder when $N$ is divided by $1000$.