
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 defined by equations $x^2+y^2-12x+16y-160=0$ and $x^2+y^2+12x+16y-36=0$, respectively. Find the smallest positive value of $k$ such that the line $y=kx+1$ contains the center of a circle that is externally tangent to $u_2$ and internally tangent to $u_1$. Given that $k^2=\frac{p}{q}$ where $p$ and $q$ are relatively prime integers, find $p+q$.

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