
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
Equilateral triangle \( T \) is inscribed in circle \( A \), which has a radius of \( 12 \). Circle \( B \) with radius \( 5 \) is internally tangent to circle \( A \) at one vertex of \( T \). Circles \( C \) and \( D \), both with radius \( 3 \), are internally tangent to circle \( A \) at the other two vertices of \( T \). Circles \( B \), \( C \), and \( D \) are all externally tangent to circle \( E \), which has a radius \( \dfrac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).

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