
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 $M$ is on $\overline{XY}$, $N$ is on $\overline{YZ}$, and $O$ is on $\overline{XZ}$. Let $XM = p \cdot XY$, $YN = q \cdot YZ$, and $ZO = r \cdot XZ$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of triangle $MNO$ to the area of triangle $XYZ$ can be written as $s/t$, where $s$ and $t$ are relatively prime positive integers. Find $s+t$.

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