
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 the right-angled sector \( AOB \), an arc \( OC \) (where \( C \) is the intersection point of this arc with arc \( AB \)) with center \( B \) and radius \( BO \) is drawn. Circle \( S_1 \) is tangent to arc \( AB \), arc \( OC \), and line \( OA \), while circle \( S_2 \) is tangent to arc \( AB \), line \( OA \), and circle \( S_1 \). Find the ratio of the radius of circle \( S_1 \) to the radius of circle \( S_2 \).

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