
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
Given an $8 \times 8$ board:

a) What is the minimum number of squares we must mark on this board so that each of its $3 \times 3$ sub-boards contains at least one marked square?

b) What is the minimum number of squares we must mark on this board so that each of its $3 \times 3$ sub-boards contains at least three marked squares?

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