ISL 2023 A7

Let $N$ be a positive integer. Prove that there exist three permutations $a_{1}, a_{2}, \ldots, a_{N}$; $b_{1}, b_{2}, \ldots, b_{N}$; and $c_{1}, c_{2}, \ldots, c_{N}$ of $1,2, \ldots, N$ such that $$\left|\sqrt{a_{k}}+\sqrt{b_{k}}+\sqrt{c_{k}}-2 \sqrt{N}\right|<2023$$ for every $k=1,2, \ldots, N$. (China)