
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
Circles $A_1$ and $A_2$ are externally tangent, and both are internally tangent to a larger circle $A_3$. The radii of $A_1$ and $A_2$ are 3 and 9, respectively, and the centers of the three circles are collinear. A chord of $A_3$ is also a common external tangent of $A_1$ and $A_2.$ Find the length of the chord in the form $\frac{m\sqrt{n}}{p}$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime.

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