
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 right triangle $ABC$, $\angle C = 60^\circ$. Let $C$ be the center of a circle with radius $BC$, and let the circle intersect $AC$ at point $D$. Connect $BD$. The arc $\overparen{BD}$ and chord $BD$ divide $\triangle ABC$ into three parts. Find the ratio of the areas of these three parts. $S_1 : S_2 : S_3 = \quad$.

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