
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
(Ⅰ) Given that the central angle $\alpha$ of sector OAB is $120^\circ$ and the radius $r=6$, find the length of arc AB and the area of the sector;  
(Ⅱ) Given that the perimeter of a sector is 20cm, what should be the central angle for it to have the maximum area, and what is the maximum area?

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