
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 \(C_1\) and \(C_2\) are externally tangent, and they are both internally tangent to circle \(C_3\). The radii of \(C_1\) and \(C_2\) are 5 and 13, respectively, and the centers of the three circles are all collinear. A chord of \(C_3\) is also a common external tangent of \(C_1\) and \(C_2\). Given that the length of the chord is \(\frac{m\sqrt{n}}{p}\) 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, find \(m+n+p\).

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