
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 $A O B$, an arc $O C$ (with $C$ being the intersection point of this arc with the arc $A B$) of radius $B O$ is drawn from point $B$ as its center. Circle $\omega$ is tangent to arc $A B$, arc $O C$, and line $O A$, while circle $\omega'$ is tangent to arc $O C$, line $O A$, and circle $\omega$. Find the ratio of the radius of circle $\omega$ to the radius of circle $\omega'$.

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