
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 $α$ and a radius of $r$ of the circle it belongs to:  
$(1)$ If $α=120^{\circ}$ and $r=6$, find the length of the arc of the sector.  
$(2)$ If the perimeter of the sector is $24$, what is the value of $α$ in radians when the area $S$ of the sector is maximized? And find the maximum area.

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