
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
Suppose there are 9 boxes arranged in a $3 \times 3$ square and an ample supply of black and white stones. Two players, A and B, play a stone-placing game where each player takes turns placing 3 stones in three boxes within the same row or column. The stones placed can be of any color. No mixed-color stones are allowed in any box. If a player places a white stone in a box containing black stones, one white stone and one black stone are simultaneously removed from the box. The game ends when the central and four corner boxes each have one black stone, and all other boxes are empty. At some stage in the game, if there are $x$ boxes each containing one black stone while all other boxes are empty, find all possible values of $x$.

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