
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
The plane is divided into equilateral triangles of side length 1. Consider an equilateral triangle of side length \( n \) whose sides lie on the grid lines. There is a tile on each grid point on the edge and inside this triangle. In one turn, a unit triangle is selected which has a checker on exactly 2 corners. The two stones are removed and a new stone is placed on the third corner. For which \( n \) is it possible that only one piece remains after a finite number of moves?

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