
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
In the equilateral triangle \( \triangle ABC \), a ray is drawn from vertex \( A \) towards the interior of the triangle and it reaches one side. When the ray hits a side, it reflects according to the law of reflection, meaning if it hits with an angle \( \alpha \), it leaves with an angle of \( 180^\circ - \alpha \). After \( n \) reflections, the ray returns to point \( A \) without passing through the other two vertices. Find all possible values of \( n \).

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