
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 $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect at two points, one of which is $(10, 8)$, and the product of their radii is $50$. The y-axis and the line $y = nx$ (where $n > 0$) are tangent to both circles. It is given that $n$ can be expressed in the form $d\sqrt{e}/f$, where $d$, $e$, and $f$ are positive integers, $e$ is not divisible by the square of any prime, and $d$ and $f$ are relatively prime. Find $d + e + f$.

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