
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
Three circles \( C_1, C_2, C_3 \) are arranged such that \( C_1 \) and \( C_2 \) are externally tangent to each other and both internally tangent to a larger circle \( C_3 \). The radii of \( C_1 \) and \( C_2 \) are 6 and 8, respectively, and the centers of all three circles are collinear. A chord of \( C_3 \), which is also a common external tangent of \( C_1 \) and \( C_2 \), intersects \( C_3 \). Given this arrangement, determine the length of the chord, and express your answer in simplified form as \( \frac{m\sqrt{n}}{p} \) where \( m, n, \) and \( p \) are positive integers.

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