
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 $\alpha$ and a radius of $R$, and the arc length is $l$.

(1) If $\alpha=60^\circ$ and $R=10\text{cm}$, find the arc length $l$;  
(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 the sector to be maximized;  
(3) If $\alpha= \frac {\pi}{3}$ and $R=2\text{cm}$, find the area of the segment of the circle formed by the arc.

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