
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 $PQR$, $PQ = 12$, $QR = 16$, and $PR = 20$. Point $X$ is on $\overline{PQ}$, $Y$ is on $\overline{QR}$, and $Z$ is on $\overline{PR}$. Let $PX = u \cdot PQ$, $QY = v \cdot QR$, and $RZ = w \cdot PR$, where $u$, $v$, and $w$ are positive and satisfy $u+v+w=3/4$ and $u^2+v^2+w^2=1/2$. The ratio of the area of triangle $XYZ$ to the area of triangle $PQR$ 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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