Find the maximal value of $c>0$ such that for any $n\ge 1$, and for any $n$ real numbers $x_1, \cdots, x_n$ there exists real numbers $a ,b$ such that $$\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024}$$for at least $cn$ indices $i$. Here, $x_{n+1}=x_1$ and $\{x\}$ denotes the fractional part of $x$. Proposed by Wong Jer Ren