
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 \( x^2 + y^2 + 8x - 20y - 75 = 0 \) and \( x^2 + y^2 - 8x - 20y + 125 = 0 \), respectively. Let \( m \) be the smallest positive value of \( a \) for which the line \( y = ax \) contains the center of a circle that is externally tangent to \( w_2 \) and internally 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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