
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
In triangle $DEF$ with right angle at $F$, $\angle DFE < 45^\circ$ and $DE = 5$. Point $Q$ on $\overline{DE}$ is chosen such that $\angle DQF = 2\angle DFQ$ and $FQ = 2$. Find the ratio $\frac{DQ}{EQ}$ in the form $u + v\sqrt{w}$, where $u$, $v$, and $w$ are positive integers and $w$ is prime.

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