
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
Dominoes of size $1 \times 2$ are placed without overlapping on a chessboard of size $8 \times 8$. The dominoes can extend beyond the borders of the board, but the center of each domino must lie strictly within the board (not on the boundary). Place on the board

a) at least 40 dominoes;

b) at least 41 dominoes;

c) more than 41 dominoes.

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