
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
Consider a parallelepiped $PQRS TUVW$ constructed using vectors $\overrightarrow{PQ}, \overrightarrow{PR}, \overrightarrow{PT}$ originating from point $P$. The vectors have magnitudes $\|\overrightarrow{PQ}\| = a$, $\|\overrightarrow{PR}\| = b$, and $\|\overrightarrow{PT}\| = c$. The vectors satisfy the condition: $\overrightarrow{PQ} \cdot \overrightarrow{PR} = x$, $\overrightarrow{PQ} \cdot \overrightarrow{PT} = y$, and $\overrightarrow{PR} \cdot \overrightarrow{PT} = z$. Compute the ratio 
\[
\frac{QT^2 + RV^2 + PS^2 + UW^2}{PQ^2 + PR^2 + PT^2}.
\]

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