Question 192
Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v} = (v_1, \ldots, v_{2022})$ and $\mathbf{w} = (w_1, \ldots, w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple:
$$
\begin{aligned}
& \mathbf{v} + \mathbf{w} = (v_1 + w_1, \ldots, v_{2022} + w_{2022}) \\
& \mathbf{v} \vee \mathbf{w} = (\max(v_1, w_1), \ldots, \max(v_{2022}, w_{2022}))
\end{aligned}
$$
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?
$$
\begin{aligned}
& \mathbf{v} + \mathbf{w} = (v_1 + w_1, \ldots, v_{2022} + w_{2022}) \\
& \mathbf{v} \vee \mathbf{w} = (\max(v_1, w_1), \ldots, \max(v_{2022}, w_{2022}))
\end{aligned}
$$
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?