
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
Two circles with centers at $(3,7)$ and $(10,12)$ have radii $3$ and $7$, respectively. Find the y-intercept $b$ of the equation of a common external tangent line to the circles. Assume the line has a positive slope and can be expressed in the form $y = mx + b$.
A) $\frac{900}{119}$  
B) $\frac{910}{119}$  
C) $\frac{912}{119}$  
D) $\frac{914}{119}$  
E) $\frac{925}{119}$

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