
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 $PQR$, $PQ=14$, $QR=16$, and $RP=18$. Point $S$ is on $\overline{PQ}$, $T$ is on $\overline{QR}$, and $U$ is on $\overline{RP}$. Let $PS=x\cdot PQ$, $QT=y\cdot QR$, and $RU=z\cdot RP$, where $x$, $y$, and $z$ are positive and satisfy $x+y+z=3/4$ and $x^2+y^2+z^2=3/7$. Find the ratio of the area of triangle $STU$ to the area of triangle $PQR$ and express it in simplest fractional form.

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