
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
Given integers \(m \geq 2, n \geq 3\), let the set
\[ 
S = \{ (a, b) \mid a \in \{1, 2, \ldots, m\}, b \in \{1, 2, \ldots, n\} \}
\]
and \(A\) be a subset of \(S\). If there do not exist positive integers \(x_{1}, x_{2}, y_{1}, y_{2}, y_{3}\) such that \(x_{1} < x_{2}\), \(y_{1} < y_{2} < y_{3}\), and \((x_{1}, y_{1}), (x_{1}, y_{2}), (x_{1}, y_{3}), (x_{2}, y_{2}) \in A\), find the maximum number of elements in the set \(A\).

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