
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
An equilateral triangle $U$ is inscribed in a circle $F$ with a radius of $15$. Circle $G$ with a radius of $4$ is internally tangent to circle $F$ at one vertex of $U$. Circles $H$ and $I$, both with a radius of $3$, are internally tangent to circle $F$ at the other two vertices of $U$. Circle $J$, with a radius of $1$, is tangent to circle $F$ halfway between two vertices of $U$. Circles $G$, $H$, $I$, and $J$ are all externally tangent to circle $K$, which has radius $r$. Find the value of $20r$.

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