
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
Triangle $DEF$ has $DE = 600$ and $EF = 400$. Points $Q$ and $R$ are located on $\overline{DE}$ and $\overline{DF}$ respectively such that $DQ = QE$, and $\overline{ER}$ is the angle bisector of angle $E$. Let $S$ be the point of intersection of $\overline{FQ}$ and $\overline{ER}$, and let $N$ be the point on line $FQ$ for which $Q$ is the midpoint of $\overline{SN}$. If $DN = 240$, find $RS$.

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