
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
Points \( A \), \( B \), and \( C \) lie on a straight line. Segment \( AB \) is the diameter of the first circle, and segment \( BC \) is the diameter of the second circle. A line passing through point \( A \) intersects the first circle at point \( D \) and is tangent to the second circle at point \( E \). Given \( BD = 9 \) and \( BE = 12 \), find the radii of the circles.

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