
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 colored black so that in every L-shaped corner of three squares, there is at least one uncolored square?

b) What is the minimum number of squares on an $8 \times 8$ board that must be colored black so that in every L-shaped corner of three squares, there is at least one black square?

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