Christmas Contest - Individual Round - Problem 18

Let $\mathcal{S}$ be the set of all functions $f: \mathbb{Z} \longrightarrow \mathbb{R}$ such that $f(x+2024) - 2024 \le f(x) \le f(x+2277) - 2277$ for all $x \in \mathbb{Z}$, and $f(1) = 1$. Find the minimum integer $k > 1$ such that $f(k)$ is the same across all $f \in \mathcal{S}$.