
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
Given points $P$ and $Q$ on the ellipse $\frac{x^2}{2} + y^2 = 1$, such that $PF\_2 \perp QF\_2$, where $F\_1$ and $F\_2$ are the left and right foci, respectively.

1. Find the minimum value of $| \overrightarrow{PF\_1} + \overrightarrow{PF\_2} |$.
2. If $( \overrightarrow{PF\_1} + \overrightarrow{PF\_2} ) \perp ( \overrightarrow{QF\_1} + \overrightarrow{QF\_2} )$, let the slope of line $PQ$ be $k$, find the value of $k^2$.

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