MathNet 2020 — Math Olympiad Vault — Part 02 Question 71

Let $S$ be a set of integers. Call $S$ beautiful if it contains every integer of form $2^{a}-2^{b}$, where $a$ and $b$ are positive integers. Call $S$ strong if, for every nonconstant polynomial $P(X)$ with coefficients in $S$, all integer roots of $P(X)$ also belong to $S$.
Find all sets that are both beautiful and strong.