
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 central angle $\alpha$ and radius $R$ of the circle:
(1) If $\alpha = 60^\circ$ and $R = 10$ cm, find the length of the arc and the area of the segment formed by the arc.
(2) If the perimeter of the sector is 12 cm, at what value of $\alpha$ in radians does the sector have the maximum area? Also, find the maximum area.

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