
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 $R$.

$(1)$ If $α=60^{\circ}$ and $R=10cm$, find the arc length and the area of the segment formed by the arc.

$(2)$ If the perimeter of the sector is a constant value $C (C > 0)$, what should be the value of $α$ in radians for the sector to have the maximum area?

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