
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 a  $12\times 12$  grid, colour each unit square with either black or white, such that there is at least one black unit square in any  $3\times 4$  and  $4\times 3$  rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.

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