
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
Vasya cut a triangle out of cardboard and numbered its vertices with the numbers 1, 2, and 3. It turns out that if Vasya rotates his triangle 15 times clockwise around the vertex numbered 1 by an angle equal to the angle at this vertex, the triangle returns to its original position. If Vasya rotates his triangle 6 times clockwise around the vertex numbered 2 by an angle equal to the angle at this vertex, the triangle returns to its original position. Vasya claims that if he rotates his triangle \( n \) times around the vertex numbered 3 by an angle equal to the angle at this vertex, the triangle will return to its original position. What is the minimal \( n \) that Vasya could name so that his statement is true for at least some cardboard triangle?


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