
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 $PQRS$ formed by vectors $\overrightarrow{PQ},$ $\overrightarrow{PR},$ and $\overrightarrow{PS}$ in three-dimensional space. Compute the following ratio:
\[
\frac{\overrightarrow{PQ}^2 + \overrightarrow{PR}^2 + \overrightarrow{PS}^2 + \overrightarrow{QR}^2 + \overrightarrow{QS}^2 + \overrightarrow{RS}^2}{\overrightarrow{PQ}^2 + \overrightarrow{PR}^2 + \overrightarrow{PS}^2}.
\]

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