2022 IMO Shortlist C7

Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=\left(v_{1}, \ldots, v_{2022}\right)$ and $\mathbf{w}=\left(w_{1}, \ldots, w_{2022}\right)$ that she has already written, and apply one of the following operations to obtain a new tuple: $$\begin{aligned} & \mathbf{v}+\mathbf{w}=\left(v_{1}+w_{1}, \ldots, v_{2022}+w_{2022}\right) \\ & \mathbf{v} \vee \mathbf{w}=\left(\max \left(v_{1}, w_{1}\right), \ldots, \max \left(v_{2022}, w_{2022}\right)\right) \end{aligned}$$ and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued 2022-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote? (Czech Republic)

Lucy 首先在黑板上写下 $s$ 整数值 2022 元组。完成此操作后，她可以采用她已编写的任意两个(不一定不同)元组 $\mathbf{v}=\left(v_{1}, \ldots, v_{2022}\right)$ 和 $\mathbf{w}=\left(w_{1}, \ldots, w_{2022}\right)$，并应用以下操作之一来获取新元组： \begin{对齐} & \mathbf{v}+\mathbf{w}=\left(v_{1}+w_{1}, \ldots, v_{2022}+w_{2022}\right) \\ & \mathbf{v} \vee \mathbf{w}=\left(\max \left(v_{1}, w_{1}\right), \ldots, \max \left(v_{2022}, w_{2022}\right)\right) \end{aligned} 然后将此元组写在黑板上。事实证明，通过这种方式，Lucy 可以在有限的步骤之后在黑板上写出任何整数值的 2022 元组。她最初编写的元组的最小可能数量 $s$ 是多少？ (捷克共和国)