
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 an acute triangle \( ABC \), \(\angle A = 30^\circ\). Taking \( BC \) as the diameter, a circle is drawn which intersects \( AB \) and \( AC \) at points \( D \) and \( E \) respectively. Connecting \( DE \) divides triangle \( ABC \) into triangle \( ADE \) and quadrilateral \( BDEC \). Suppose their areas are \( S_1 \) and \( S_2 \) respectively. Find the ratio \( S_1 : S_2 \).

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