
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 the ellipse \(\frac{x^2}{2} + y^2 = 1\), two mutually perpendicular lines pass through the fixed point \(C(1,0)\) and 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 slopes of the lines on which points \(P\) and \(Q\) lie.

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