
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 $C_1$ and $C_2$ with radii 6 and 12 respectively, are externally tangent to each other and internally tangent to a larger circle $C_3$. All the centers of these circles are collinear. There is a chord of $C_3$, which is also a common external tangent to $C_1$ and $C_2$. Find the length of this 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. Compute $m+n+p$.

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