
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  $8\times 8$  board is given. Seven out of  $64$  unit squares are painted black. Suppose that there exists a positive  $k$  such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area  $k$  containing no black squares. Find the maximum value of  $k$ . 


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