
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 $DEF$ has each side equal to $9$. A circle centered at $Q$ is tangent to side $DE$ at $D$ and passes through $F$. Another circle, centered at $R$, is tangent to side $DF$ at $F$ and passes through $E$. Find the magnitude of segment $QR$.
A) $12\sqrt{3}$
B) $9\sqrt{3}$
C) $15$
D) $18$
E) $9$

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