
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 \( (8,4) \), and the product of the radii is 50. The x-axis, the y-axis, and the line \( y = mx \), where \( m > 0 \), are tangent to both circles. Find the value of \( m \) which can be expressed in the form \( \frac{a\sqrt{b}}{c} \), where \( a, b, \) and \( c \) are positive integers, \( b \) is 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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