Question 107
Lucile has written $s$ 2023-tuples of integers on the board. She then allows herself operations of the following form: she chooses two 2023-tuples $\mathbf{u}=\left(u_{1}, u_{2}, \ldots, u_{2023}\right)$ and $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{2023}\right)$, not necessarily distinct, among those she has already written, and then also writes on the board the two 2023-tuples $\mathbf{u}+\mathbf{v}=\left(u_{1}+v_{1}, \ldots, u_{2023}+v_{2023}\right)$ and $\max (\mathbf{u}, \mathbf{v})=\left(\max \left(u_{1}, v_{1}\right), \ldots, \max \left(u_{2023}, v_{2023}\right)\right)$.
For which values of $s$ can Lucile, by choosing wisely the $s$ 2023-tuples she wrote initially and by repeating the operations above, write any 2023-tuple of integers?
For which values of $s$ can Lucile, by choosing wisely the $s$ 2023-tuples she wrote initially and by repeating the operations above, write any 2023-tuple of integers?