Problem of the Month
Problem 4: A Polynomial Sandwich

Let $a$, $b$, $c$, and $d$ be rational numbers and $f(x)=a{x}^3+b{x}^2+cx+d$. Suppose $f(n)$ is an integer whenever $n$ is an integer and that $${\displaystyle \frac{1}{3}}{n}^3-n-{\displaystyle \frac{2}{3}}\le f(n)\le {\displaystyle \frac{1}{3}}{n}^3+n^2+2n+{\displaystyle \frac{4}{3}}$$ for every integer $n$ with the possible exception of $n=-2$.

1. Show that $a={\displaystyle \frac{1}{3}}$.
2. Find $f({10}^{2025})-f({10}^{2025}-1)$.