
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 643 square grid, there are only two types of cells: mine cells and number cells. Mine cells do not contain numbers. A number cell is filled with a number \( m \) if and only if there are \( m \) mine cells around it (8 cells around it, or 5 cells if on the border, or 3 cells if in the corner).

(1) Given a \((2n+1) \times (2n+1)\) grid containing \( n^2 + 1 \) mine cells, determine the maximum sum of numbers in all the numbered cells.

(2) Find the maximum \( k \) such that, in any \( n \times n \) grid containing \( k \) mine cells, there is at least one numbered cell with a number 0. Express the result as a function of \( n \).

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