
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 \(C: \frac{x^{2}}{2} + y^{2} = 1\) with foci \(F_{1}\) and \(F_{2}\), and a point \(P(x_{0}, y_{0})\) satisfying \(0 < \frac{x_{0}^{2}}{2} + y_{0}^{2} \leq 1\), find the range of the value \(\left|P F_{1}\right| + \left|P F_{2}\right|\).

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