
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 be a board with  $2007 \times 2007$  cells, we colour with black  $P$  cells such that:
 
 $\bullet$  there are no 3 colored cells that form a L-trinomes in any of its 4 orientations

Find the minimum value of  $P$ , such that when you colour one cell more, this configuration can't keep the condition above.

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