
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 the central angle of a sector of a circle is $\alpha$ and the radius of the circle is $r$.

(1) If $\alpha = 120^{\circ}$ and $r = 6$, find the length of the arc of the sector.

(2) If the perimeter of the sector is 24, for what value of $\alpha$ (in radians) is the area of the sector ($S$) maximized, and what is the maximum value?

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