
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
Let $u_1$ and $u_2$ denote the circles $x^2+y^2+8x-18y-75=0$ and $x^2 +y^2-8x-18y+135=0,$ respectively. Let $n$ be the smallest positive value of $b$ for which the line $y=bx$ contains the center of a circle that is externally tangent to $u_2$ and internally tangent to $u_1.$ Given that $n^2=\frac rs,$ where $r$ and $s$ are relatively prime integers, find $r+s.$

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