
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 \( k \). On a coordinate plane, initially \( N \) cells are marked. We call a set of cells \( A \) a cross if all cells in the same vertical or horizontal line as \( A \) are included. If there are at least \( k \) other marked cells in the cross of an unmarked cell \( A \), then cell \( A \) can also be marked. It turns out that any cell on the plane can be marked through a series of such actions. What is the smallest possible value of \( N \) for which this could happen?

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