
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
Let  $ABC$  be an equilateral triangle. From the vertex  $A$  we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle  $\alpha$ , it leaves with a directed angle  $180^{\circ}-\alpha$ . After  $n$  bounces, the ray returns to  $A$  without ever landing on any of the other two vertices. Find all possible values of  $n$ .

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