
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
Let  $n$  be a positive integer. Find all integers  $k$  among  $1,2,\dots,2n^2$  which satisfy the following condition:
・There is a  $2n\times 2n$  grid of square unit cells. When  $k$  different cells are painted black while the other cells are painted white, the minimum possible number of  $2\times 2$  squares that contain both black and white cells is  $2n-1$ .

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