
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
The radius \( r \) of a circle inscribed within three mutually externally tangent circles of radii \( a = 3 \), \( b = 6 \), and \( c = 9 \) is given by the formula
\[
\frac{1}{r} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 2 \sqrt{\frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc}}.
\]
Calculate the value of \( r \).

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