
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
There are three semicircular protractors with radii of $1 \text{ cm}$, $3 \text{ cm}$, and $5 \text{ cm}$, whose centers coincide at point $O$, and their diameters also align on a straight line as shown in the diagram. Denote the areas of the shaded sectors in the semicircles with radii $1 \text{ cm}$, $3 \text{ cm}$, and $5 \text{ cm}$ as $S_{\text{A}}$, $S_{\text{B}}$, and $S_{\text{C}}$ respectively. Find the ratio $S_{\text{A}} : S_{\text{B}} : S_{\text{C}}$.

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