
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
A regular hexagon inscribed in a circle has each side 5 units long. Find both:
1. The area of the sector formed by the angle subtended by one side of the hexagon at the center of the circle.
2. The area of one equilateral triangle formed by three consecutive vertices of the hexagon.
A) $\frac{25\pi}{6}$, $\frac{25\sqrt{3}}{4}$
B) $\frac{25\pi}{3}$, $\frac{50\sqrt{3}}{4}$
C) $\frac{12.5\pi}{3}$, $\frac{12.5\sqrt{3}}{2}$
D) $12.5\pi$, $12.5\sqrt{3}$

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