
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
In an $8 \times 12$ table, some $N$ cells are black, while the rest are white. In one operation, it is allowed to paint three cells that form a three-cell corner to white (some of them might have been white before repainting). It turns out that it is impossible to make the entire table completely white in fewer than 25 such operations. Find the smallest possible value of $N$.

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