
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 an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(0,1)$, and the length of the major axis is $\sqrt{2}$ times the focal length. A line passing through the left focus $F$ of the ellipse $C$ intersects the ellipse at points $A$ and $B$, with $O$ being the origin.  
(Ⅰ) Find the standard equation of the ellipse $C$;  
(Ⅱ) If the line $AB$ is perpendicular to the $x$-axis, determine the position relationship between point $O$ and the circle with diameter $AB$, and explain the reason;  
(Ⅲ) If point $O$ is inside the circle with diameter $AB$, find the range of values for the slope $k$ of line $AB$.

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