
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 $ABC$ has $AB=24$, $AC=26$, and $BC=30$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the centroid of triangle $ABC$. Find $DE$ expressed as a fraction in lowest terms, and calculate $m+n$ where $DE=\frac{m}{n}$ and $m$ and $n$ are relatively prime positive integers.

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