
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
We have an infinite grid, and we need to write a positive integer in each cell with the following conditions. The number $n$ should appear exactly $n$ times (for example, the number 1 appears once, the number 2 appears twice, etc.), and the difference between numbers in any two adjacent cells (cells sharing a common side) should be less than a given $k$. What is the smallest integer $k$ for which this arrangement is possible?

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