
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 $ABC$, $AB = 12$, $BC = 16$, and $CA = 20$. Point $D$ is on $\overline{AB}$, $E$ is on $\overline{BC}$, and $F$ is on $\overline{CA}$. Let $AD=p\cdot AB$, $BE=q\cdot BC$, and $CF=r\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=1$ and $p^2+q^2+r^2=1/2$. Calculate the ratio of the area of triangle $DEF$ to the area of triangle $ABC$. If this ratio can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.

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