
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
A $m \times n (m < n)$ grid is divided into $mn$ unit squares. Initially, $N$ squares are painted black. We then paint other squares black according to the following rule: if an empty square has at least two neighboring black squares (sharing a common edge), then paint that square black. Determine the minimum value of $N$ such that, after a finite number of steps, all squares in the grid are painted black.

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