Let $f(x)=-x^2+ax+b$, where $a,b\in\mathbb{R}$. Suppose there exist distinct integers $m,n$ such that $f(m)=-n^2$ and $f(n)=-m^2$.

Prove that there are infinitely many pairs of integers $x,y$ such that $f(x)=-y^2$ and $f(y)=-x^2$.

[Source: Russian competition problem]