KöMaL 2017 B4914

Let $p(x)$ be a polynomial of integer coefficients such that there exist $4$ distinct integers $a$, $b$, $c$, $d$ with $p(a)=p(b)=p(c)=p(d)=2000$. $a)$ Prove that there is no integer $x_0$ for which $p(x_0)=2017$. $b)$ Find a polynomial $p(x)$ that meets the above requirement, and find an integer $x_1$ for which $p(x_1)=2018$.