US-TST 2023 Problem 3

Consider pairs $(f, g)$ of functions from the set of nonnegative integers to itself such that - $f(0) \geq f(1) \geq f(2) \geq \cdots \geq f(300) \geq 0 ;$ - $f(0)+f(1)+f(2)+\cdots+f(300) \leq 300$; - for any 20 nonnegative integers $n_{1}, n_{2}, \ldots, n_{20}$, not necessarily distinct, we have $$g\left(n_{1}+n_{2}+\cdots+n_{20}\right) \leq f\left(n_{1}\right)+f\left(n_{2}\right)+\cdots+f\left(n_{20}\right) .$$ Determine the maximum possible value of $g(0)+g(1)+\cdots+g(6000)$ over all such pairs of functions.