
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
For positive integers \( m \) and \( n \), let \( A(m, n) \) be the set of \( 2 \times mn \) matrices \( M \) with entries from the set \( \{1, 2, \ldots, m\} \) such that:
- Each of the numbers \( 1, 2, \ldots, m \) appears exactly \( 2n \) times;
- \( M_{1,1} \leq M_{1,2} \leq \cdots \leq M_{1, mn} \) and \( M_{2,1} \leq M_{2,2} \leq \cdots \leq M_{2, mn} \); and
- \( M_{1, j} < M_{2, j} \) for \( j = 1, 2, \ldots, mn \).

Determine \( |A(m, n)| \).

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