
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 area of an isosceles triangle is \( S \), and the angle between the altitude to the lateral side and the base is \( \alpha \). Find the radius of the circle inscribed in the triangle.

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