
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 $c_1$ and $c_2$ denote the circles given by $x^2 + y^2 + 8x - 20y - 75 = 0$ and $x^2 + y^2 - 8x - 20y + 175 = 0$, 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 $c_2$ and internally tangent to $c_1$. Find $n^2$, expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers, then find $p+q$.

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