US-TST 2018 Problem 6

Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\ell$ in the plane which is horizontal, vertical, or has slope +1 or -1 , she tells Bob the number of points of $S$ that lie on $\ell$. Bob wins if he can then determine the set $S$. Prove that if Alice picks $S$ to be of the form $$S=\left\{(x, y) \in \mathbb{Z}^{2} \mid m \leq x^{2}+y^{2} \leq n\right\}$$ for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.)