
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 \( m \geq 3 \) and \( n \geq 3 \), 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 \( x_1, x_2, x_3, y_1, y_2, y_3 \) such that \( x_1 < x_2 < x_3 \), \( y_1 < y_2 < y_3 \), and \((x_1, y_2)\), \((x_2, y_1)\), \((x_2, y_2)\), \((x_2, y_3)\), \((x_3, y_2) \in A\), find the maximum number of elements in the set \( A \).

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