
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 $A$, having a radius of $10$. Circle $B$ with a radius of $4$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, each with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. All three circles $B$, $C$, and $D$ are externally tangent to circle $E$, which has radius $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Determine the value of $p+q$.

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