
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
Divide an \( m \)-by-\( n \) rectangle into \( m n \) nonoverlapping 1-by-1 squares. A polyomino of this rectangle is a subset of these unit squares such that for any two unit squares \( S \) and \( T \) in the polyomino, either

(1) \( S \) and \( T \) share an edge, or

(2) there exists a positive integer \( n \) such that the polyomino contains unit squares \( S_1, S_2, S_3, \ldots, S_n \) such that \( S \) and \( S_1 \) share an edge, \( S_n \) and \( T \) share an edge, and for all positive integers \( k < n \), \( S_k \) and \( S_{k+1} \) share an edge.

We say a polyomino of a given rectangle spans the rectangle if for each of the four edges of the rectangle the polyomino contains a square whose edge lies on it.

What is the minimum number of unit squares a polyomino can have if it spans a 128-by-343 rectangle?

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