
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 $m \times n$ chessboard (consisting of $m$ rows and $n$ columns of squares, where $m \geq 3$ and $n \geq 3$), each square is colored either red or blue. If two adjacent squares (sharing a common edge) have different colors, they are called a "standard pair." Let $S$ denote the number of "standard pairs" on the chessboard. Determine whether $S$ is odd or even based on the coloring of certain squares. Under what conditions is $S$ odd? Under what conditions is $S$ even? Explain your reasoning.

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