
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 $(10,8)$. The product of the radii is $75$. The x-axis and the line $y = 2mx$, where $m > 0$, are tangent to both circles. It is stated that $m$ can be expressed as $a\sqrt{b}/c$, where $a$, $b$, and $c$ are positive integers with $b$ not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Determine $a+b+c$.

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