
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 a parallelepiped $PQRS$ constructed from vectors $\overrightarrow{PQ}, \overrightarrow{PR},$ and $\overrightarrow{PS},$ suppose $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ are orthogonal and of equal magnitude, and $\overrightarrow{PS}$ is twice the magnitude of $\overrightarrow{PQ}$. Calculate the following ratio:
\[
\frac{QR^2 + PS^2}{PQ^2 + PR^2 + PS^2}.
\]

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