
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 $F_1F_2$, and a point $P(x_0,y_0)$ such that $0 < \frac{x_0^2}{2} + y_0^2 < 1$. Determine the range of values for $|PF_1| + |PF_2|$ and the number of common points between the line $\frac{x_0x}{2} + y_0y = 1$ and the ellipse $C$.

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