
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 \( k \). On a grid plane, \( N \) cells have been pre-marked. The set of all cells in the same row or column as cell \( A \) is called the "cross" of cell \( A \). If \( k \) other cells in the cross of an unmarked cell \( A \) are already marked, then cell \( A \) can also be marked. It is known that after a series of such operations, all cells in the grid plane can be marked. What is the smallest \( N \) for which this situation can occur?

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