
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 $PQRS$ defined by vectors $\overrightarrow{PQ},$ $\overrightarrow{PR},$ and $\overrightarrow{PS},$ where $\overrightarrow{PQ} = 2\mathbf{i} + \mathbf{j}, \overrightarrow{PR} = \mathbf{j} + 2\mathbf{k}, \overrightarrow{PS} = 2\mathbf{i} + \mathbf{k}$. Compute the following ratio:
\[
\frac{QR^2 + PS^2 + QT^2 + RS^2}{PQ^2 + PR^2 + PS^2}.
\]

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