
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 game is played as follows: Initially, \( n^2 \) pieces are arranged in an \( n \times n \) block of adjoining squares, one piece on each square. A move in the game consists of a piece jumping in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece that has been jumped over is removed. Find the values of \( n \) for which the game can end with only one piece remaining on the board.

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