
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
Two circles are externally tangent, one centered at \( A \) with a radius of \( 2 \) units and another centered at \( B \) with a radius of \( 3 \) units. The line segment connecting the centers \( AB \) is perpendicular to the x-axis, and the distance between the centers is \( 6 \) units. A third circle is tangent to the first two circles and to one of their common external tangents. Find the radius of the third circle.

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