
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  $Q$  be a  $(2n+1) \times (2n+1)$  board. Some of its cells are colored black in such a way that every  $2 \times 2$  board of  $Q$  has at most  $2$  black cells. Find the maximum amount of black cells that the board may have.

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