
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

From an isosceles triangle with a vertex angle \(\alpha\) and an area of 1, the largest possible circle is cut out. Then, from this circle, the largest possible triangle similar to the original isosceles triangle is cut out. What is the maximum and minimum value of the area \(S(\alpha)\) of the resulting triangle for \(60^{\circ} \leq \alpha \leq 120^{\circ}\)?

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