
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 the positive integer  $m \geq 2$ ,  $n \geq 3$ . Define the following set  $$ S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}. $$  Let  $A$  be a subset of  $S$ . If there does 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  $$ (x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A. $$  Determine the largest possible number of elements in  $A$ .

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