
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 any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *score* is the sum of all numbers recorded this way. Deyuan shades each square in a blank   $n\times n$  grid with probability  $k$ ; he notices that the expected value of the score of the resulting grid is equal to  $k$ , too! Given that  $k > 0.9999$ , find the minimum possible value of  $n$ .

*Proposed by Andrew Wu*

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