
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 positive integers \( m \) and \( n \) where \( m < 2001 \) and \( n < 2002 \), there are 2001 × 2002 distinct real numbers. These numbers are placed into a 2001 × 2002 rectangular chessboard (the chessboard has 2001 rows and 2002 columns) such that each cell contains one number, and each number is placed in one cell. A cell is called a "bad cell" if the number in the cell is less than at least \( m \) numbers in its column and less than at least \( n \) numbers in its row. For each arrangement, let \( S \) be the number of "bad cells." Find the minimum value of \( S \).

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