
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
The diagonal \(AC\) of a convex quadrilateral \(ABCD\) is the diameter of the circumcircle. Find the ratio of the areas of triangles \(ABC\) and \(ACD\) if it is known that the diagonal \(BD\) divides \(AC\) in the ratio 2:1 (counting from point \(A\)), and \(\angle BAC = 30^{\circ}\).

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