
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 an infinite sheet of square ruled paper, some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighboring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an \( m \times n \) rectangle (\( m, n > 1 \)) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?

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