CN-LEAGUE 2021 Problem 4

Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$ , if $|A| >cn$ , there exists a function $f:A\to\{1,-1\}$ satisfying $$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$