
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

a) What is the maximum number of squares on an $8 \times 8$ board that can be painted black such that in each corner of three squares, there is at least one unpainted square?

b) What is the minimum number of squares on an $8 \times 8$ board that need to be painted black such that in each corner of three squares, there is at least one black square?

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