
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 $AB=13$, $AC=14$, and $BC=15$. Points $F$ and $G$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{FG}$ is parallel to $\overline{BC}$ and contains the centroid of triangle $ABC$. Find $FG$ in reduced form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?

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