
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]


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Given the ellipse \(\frac{x^{2}}{2} + y^{2} = 1\), two mutually perpendicular moving lines passing through point \(C(1,0)\) intersect the ellipse at points \(P\) and \(Q\). \(F_1\) and \(F_2\) are the left and right foci respectively, and \(O\) is the origin.

1. Find the minimum value of the vector \(\left|\overrightarrow{P F_{1}} + \overrightarrow{P F_{2}}\right|\).
2. When the vectors \(\overrightarrow{P F_{1}} + \overrightarrow{P F_{2}}\) and \(\overrightarrow{Q F_{1}} + \overrightarrow{Q F_{2}}\) are perpendicular to each other, find the slope of the line passing through points \(P\) and \(Q\).

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