
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 $P$, which has radius $12$. Circle $Q$ with radius $4$ is internally tangent to circle $P$ at one vertex of $U$. Circles $R$ and $S$, both with radius $3$, are internally tangent to circle $P$ at the other two vertices of $U$. Circles $Q$, $R$, and $S$ are all externally tangent to circle $F$, which has 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}
user