
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
Given a sector with a central angle of \\(\alpha\\), radius \\(R\\), and arc length \\(l\\).  
\\((1)\\) If \\(\alpha = 60^\circ\\) and \\(R = 10 \, \text{cm}\\), find the arc length \\(l\\) of the sector;  
\\((2)\\) If the perimeter of the sector is \\(20 \, \text{cm}\\), what should be the value of the central angle \\(\alpha\\) in radians for the area of this sector to be maximized;  
\\((3)\\) If \\(\alpha = \frac{\pi}{3}\\) and \\(R = 2 \, \text{cm}\\), find the area of the segment of the circle defined by the arc of the sector.

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