
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 sector's area to be maximized?

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