
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 $U$ is inscribed in circle $F$, which has a radius of $15$ units. Circle $G$ with radius $5$ units is internally tangent to circle $F$ at one vertex of $U$. Circle $H$, having a radius of $3$ units, is internally tangent to circle $F$ at another vertex of $U$. Circles $I$, having a radius of $2$ units, is tangent not only to circle $F$ but also at another vertex of $U$. Circles $G$, $H$, and $I$ are externally tangent to circle $J$, with its radius being $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Calculate $p+q$.

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