
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 infinitely extendable chessboard, a game is played according to the following rules: First, $n^{2}$ pieces are placed on an $n \times n$ square grid composed of contiguous small squares, with one piece in each small square. In each step of the game, a piece is moved horizontally or vertically over an adjacent small square that has a piece to an empty small square, otherwise the move is not allowed, and then the piece that was jumped over is removed. Find all values of $n$ such that the game ultimately results in only one piece remaining on the chessboard.

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