
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

A circle with center at point \( M \) touches the sides of the angle \( AOB \) at points \( A \) and \( B \). A second circle with center at point \( N \) touches segment \( OA \), ray \( BA \), and the extension of the side of the angle \( OB \) beyond point \( O \). It is known that \( ON: OM = 12: 13 \). Find the ratio of the radii of the circles.

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