
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
Right triangle $DEF$ has side lengths $DE=9$, $EF=12$, and $DF=15$. A circle centered at $Q$ is tangent to line $DE$ at $D$ and passes through $F$. A circle centered at $R$ is tangent to line $EF$ at $E$ and passes through $D$. Find the distance $QR$.

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