
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
In triangle $ABC$, the projection of vertex $C$ on side $AB$ is denoted as $C_{1}$. We know that the sector of the circle centered at $C$ with radius $C C_{1} = m_{c}$ that lies inside the triangle has an area that is half of the area of the triangle. Determine the angles of the triangle to the nearest minute, given the following conditions: I. there is a right angle in the triangle, II. two angles are equal. How long is the area-bisecting arc? Is the area-bisecting line segment shorter if a straight line segment parallel to $AB$ is used instead of the circular arc? $(\pi \approx 3.1416)$.

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