
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 $A$ has radius $9$ units and circle $B$ has radius $3$ units. The circles are externally tangent to each other at point $C$. Segment $UV$ is the common external tangent to circle $A$ and circle $B$ at points $U$ and $V$, respectively. What is the length of segment $AV$? Express your answer in simplest radical form.

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