TR-MO 2019 Problem 5

Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$ , there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$\sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0.$$ Determine the maximum possible value of $$\sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i).$$