2016 IMO Shortlist A3

Find all integers $n \geq 3$ with the following property: for all real numbers $a_{1}, a_{2}, \ldots, a_{n}$ and $b_{1}, b_{2}, \ldots, b_{n}$ satisfying $\left|a_{k}\right|+\left|b_{k}\right|=1$ for $1 \leq k \leq n$, there exist $x_{1}, x_{2}, \ldots, x_{n}$, each of which is either -1 or 1 , such that $$\left|\sum_{k=1}^{n} x_{k} a_{k}\right|+\left|\sum_{k=1}^{n} x_{k} b_{k}\right| \leq 1$$

查找具有以下属性的所有整数 $n \geq 3$：对于所有实数 $a_{1}, a_{2}, \ldots, a_{n}$ 和 $b_{1}, b_{2}, \ldots, b_{n}$ 满足 $\left|a_{k}\right|+\left|b_{k}\right|=1$ 对于 $1 \leq k \leq n$，存在$x_{1}, x_{2}, \ldots, x_{n}$，每个都是 -1 或 1 ，这样 $$\left|\sum_{k=1}^{n} x_{k} a_{k}\right|+\left|\sum_{k=1}^{n} x_{k} b_{k}\right| \leq 1$$