
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 $XYZ$, $XY=12$, $YZ=16$, and $XZ=20$. Point $D$ is on $\overline{XY}$, $E$ is on $\overline{YZ}$, and $F$ is on $\overline{XZ}$. Let $XD=p\cdot XY$, $YE=q\cdot YZ$, and $ZF=r\cdot XZ$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=0.9$ and $p^2+q^2+r^2=0.29$. The ratio of the area of triangle $DEF$ to the area of triangle $XYZ$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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