
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 natural number \( k \), suppose \( N \) cells are initially marked on a grid. We define the "cross" of a cell \( A \) as the set of all cells lying in the same row or column as \( A \). If in the cross of an unmarked cell \( A \) there are at least \( k \) other marked cells, then cell \( A \) can also be marked.

It turned out that by repeating such actions, any cell on the grid can be marked. What is the smallest \( N \) for which this could happen?

(G. Chelnokov)

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