
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
On an infinite chessboard, a solitaire game is played as follows: At the start, we have \( n^2 \) pieces occupying \( n^2 \) squares that form a square of side \( n \). The only allowed move is a jump horizontally or vertically over an occupied square to an unoccupied one, and the piece that has been jumped over is removed. For what positive integers \( n \) can the game end with only one piece remaining on the board?

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