
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 what ratio does the point \( O \), from which all three sides of an isosceles triangle \( ABC \) are seen at the same angle \((\angle AOB = \angle BOC = \angle COA)\), divide the height of the triangle, if the angle at the base of the triangle is \( \alpha \) \((\alpha > \frac{\pi}{6})\)?

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