
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
Let \( n \geq m \) be integers.

What is the maximum number of \( 1 \times 2 \) dominoes (rotations allowed) that can be placed on an \( m \times 2n \) checkerboard such that the bottom row is completely covered by horizontal dominoes, and no two dominoes form a \( 2 \times 2 \) square?

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