
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
An $m \times n$ chessboard (composed of $m$ rows and $n$ columns of squares, where $m \geq 3$ and $n \geq 3$) is colored with two colors: red and blue. If two neighboring (sharing a common edge) squares have different colors, the pair of squares is called a "standard pair." Let $S$ be the number of "standard pairs" on the chessboard. Determine which squares' colors determine whether $S$ is odd or even, and under what circumstances $S$ is odd or even. Explain your reasoning.

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