USAMO 2022 Problem 5

A function $f: \mathbb{R}\to \mathbb{R}$ is essentially increasing if $f(s)\leq f(t)$ holds whenever $s\leq t$ are real numbers such that $f(s)\neq 0$ and $f(t)\neq 0$ .

Find the smallest integer $k$ such that for any 2022 real numbers $x_1,x_2,\ldots , x_{2022},$ there exist $k$ essentially increasing functions $f_1,\ldots, f_k$ such that $$f_1(n) + f_2(n) + \cdots + f_k(n) = x_n \text{for every } n= 1,2,\ldots 2022.$$