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:

\[\begin{aligned} \mathbf{v}+\mathbf{w} &= (v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w} &= (\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{aligned}\]

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?