
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 \( x^2+y^2-6x+8y-23=0 \) and \( x^2+y^2+6x+8y+89=0 \), respectively. Let \( m \) be the smallest positive value of \( a \) for which the line \( y = ax + 3 \) contains the center of a circle \( w_3 \) that is externally tangent to both \( w_1 \) and \( w_2 \). 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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