
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 point $C$ moves on the arc $AB$ of a circle with center $O$. The line $OC$ intersects the tangent to the circle at $B$ at point $D$. Determine the positions of $C$ in which the ratio of the area of the sector $AOC$ to the area of the triangle $BOD$ is $k = 2, 1, 1/2$. Provide the angular measurement of $\angle AOC$ in radians to an accuracy of 0.001.

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