
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
Given a positive integer \( n \), in an \( n \times n \) grid, each cell is filled with an integer. Perform the following operation on this grid: select a cell and add 1 to each of the \( 2n-1 \) numbers in that cell's row and column. Determine the largest integer \( N = N(n) \) such that no matter what the initial numbers in the grid are, you can always make at least \( N \) even numbers in the grid after a series of the described operations.

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