Find all integers $n\ge 3$ with the following property: for all real numbers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots, b_n$ satisfying $\left\vert a_k\right\rvert+\left\lvert b_k\right\rvert = 1$ for $1\le k\le n$, there exist $x_1,x_2,\dots,x_n$ each of which is either $-1$ or $1$, such that
$$\Big\lvert \sum _{k=1}^nx_ka_k \Big\rvert + \Big\lvert \sum _{k=1}^nx_kb_k\Big\rvert \le 1$$