
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 \( k \) and \( n \) be integers with \( k \geq 2 \) and \( k \leq n \leq 2k-1 \). We place some \( 1 \times k \) or \( k \times 1 \) rectangular paper pieces on an \( n \times n \) grid such that each piece covers exactly \( k \) cells, and no two pieces overlap, until no new piece can be placed. For any such \( k \) and \( n \), find the minimum possible number of pieces that can be placed to satisfy these conditions.

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