
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 the parallelepiped $PQRS TUVW$ generated by vectors $\overrightarrow{PQ},$ $\overrightarrow{PR},$ and $\overrightarrow{PT}$. Let $\overrightarrow{PQ} = 2\mathbf{i} + \mathbf{j}$, $\overrightarrow{PR} = \mathbf{i} + 2\mathbf{j} + \mathbf{k}$, and $\overrightarrow{PT} = 3\mathbf{k}$. Calculate the value of:
\[
\frac{QT^2 + RV^2 + PS^2 + UW^2}{PQ^2 + PR^2 + PT^2}.
\]

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