
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
Given right triangle $DEF$ with side lengths $DE=9$, $EF=12$, and $DF=15$, and a circle centered at $Q$ tangent to line $DE$ at $D$ and passing through $F$, and a circle centered at $R$ tangent to line $DF$ at $F$ and passing through $E$, determine the length of segment $QR$.

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