
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 chord of a circle with a radius of 1 unit is equal to the side of an inscribed equilateral triangle. The chord is then rotated $90^{\circ}$ around the center of the circle. What is the area swept by the chord during this rotation? Also, by what angle should the chord be rotated so that it sweeps exactly half of the area of the circle?

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