
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 $\triangle{DEF}$ where $DE = 18$, $EF = 28$, and $DF = 34$, let $N$ be a point on $\overline{DF}$ such that the incircles of $\triangle{DEN}$ and $\triangle{EFN}$ have equal radii. Find the ratio $\frac{DN}{NF}$ and express it as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Then find $p + q$.

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