
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 that a sector has a central angle of $\alpha$ and the radius of the circle is $R$.  
(1) If $\alpha = 60^\circ$ and $R = 10$ cm, find the arc length and the area of the sector.  
(2) If the perimeter of the sector is 12 cm, what is the value of $\alpha$ in radians when the sector has the maximum area, and what is that maximum area?

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