
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
In each square of a chessboard, there are two cockroaches. At a certain moment, each cockroach crawls to an adjacent (by side) square, and the cockroaches that were in the same square crawl to different squares. What is the maximum number of squares on the board that can remain empty after this?

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