
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
Let $k>1$ be a positive integer. Determine the smallest integer $n$ for which it is possible to color some cells black in an $n \times n$ grid such that no two black cells share a side or a corner and each row and each column contains exactly $k$ black cells.

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