
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 $RP=20$. Point $X$ is on $\overline{PQ}$, $Y$ is on $\overline{QR}$, and $Z$ is on $\overline{RP}$. Let $PX=s \cdot PQ$, $QY=t \cdot QR$, and $RZ=u \cdot RP$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=1/2$. Find the ratio of the area of triangle $XYZ$ to the area of triangle $PQR$ in the form $a/b$, where $a$ and $b$ are relatively prime positive integers, and find $a+b$.

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