
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. Initially, a  $2n \times 2n$  grid has  $k$  black cells and the rest white cells. The following two operations are allowed :
(1) If a  $2\times 2$  square has exactly three black cells, the fourth is changed to a black cell; 
(2) If there are exactly two black cells in a  $2 \times 2$  square, the black cells are changed to white and white to black.
 Find the smallest positive integer  $k$  such that for any configuration of the  $2n \times 2n$  grid with  $k$  black cells, all cells can be black after a finite number of operations.

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