
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 a radius of $5$ is internally tangent to circle $A$ at one vertex of $T$. Circle $C$ with a radius of $3$ and circle $D$ with a radius of $2$ are internally tangent to circle $A$ at the other two vertices of $T$. All three circles $B$, $C$, and $D$ are externally tangent to a smaller circle $E$, with radius $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Additionally, a circle $F$ with a radius $1$ is internally tangent to circle $A$ at the midpoint of the side opposite the vertex where circle B is tangent. Circle $F$ is also externally tangent to circle $E$. Find $m+n$.

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