
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 $|\vec{PF_1} + \vec{PF_2}|$;  
(2) If $(\vec{PF_1} + \vec{PF_2}) \perp (\vec{QF_1} + \vec{QF_2})$, and the slope of the line PQ is k, find the value of $k^2$.

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