
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 both are internally tangent to a larger circle $C_3$. The radii of $C_1$ and $C_2$ are 5 and 7, respectively, and the centers of all three circles are collinear. A tangent to $C_3$ also touches $C_2$ externally. Find the length of this tangent from the point of tangency to the circle $C_3$ to the point of tangency to the circle $C_2$. Express your answer in the form $\frac{m\sqrt{n}}{p}$ where $m, n, p$ are integers with $m$ and $p$ coprime, and $n$ is square-free.

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