
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 $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-20y-77=0$ and $x^2+y^2-10x-20y+193=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is internally tangent to $w_2$ and externally tangent to $w_1.$ Given that $m^2=\frac{p}{q},$ where $p$ and $q$ are relatively prime integers, find $p+q.$

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