
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. From this circle, the largest possible triangle similar to the original triangle is then cut out. What are the maximum and minimum values of the area $S(\alpha)$ of the resulting triangle for $60^{\circ} \leqslant \alpha \leqslant 120^{\circ}$?

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