
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
A circle $x^2+y^2=2$ intersects with the x-axis at points $F_1$ and $F_2$, and P is a point on the circle. An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ($a>b>0$) has foci at $F_1$ and $F_2$ and passes through point P.
(I) When the coordinates of point P are ($x_0$, $\frac{\sqrt{2}}{2}$) ($x_0>0$), find the value of $x_0$ and the equation of the ellipse.
(II) If a line l intersects the ellipse found in part (I) at two distinct points A and B, and point C(0, -1), $|$\vec{AC}$|$=$|$\vec{BC}$|$, find the range of values for b, the y-intercept of line l.

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