
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  $4$  and  $10$ , 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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