
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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:
\[
\begin{array}{l}
\mathbf{v}+\mathbf{w} = \left(v_{1}+w_{1}, \ldots, v_{2022}+w_{2022}\right) \\
\mathbf{v} \vee \mathbf{w} = \left(\max \left(v_{1}, w_{1}\right), \ldots, \max \left(v_{2022}, w_{2022}\right)\right)
\end{array}
\]
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?

Present the answer in LaTex format: \boxed{Your answer}
user