
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with a focal distance of $4$, and it has the same eccentricity as the ellipse $x^2 + \frac{y^2}{2} = 1$. A line $l$ with slope $k$ passes through point $M(0,1)$ and intersects ellipse $C$ at two distinct points $A$ and $B$.  
$(1)$ Find the standard equation of ellipse $C$;  
$(2)$ When the right focus $F$ of ellipse $C$ is inside the circle with diameter $AB$, find the range of values for $k$.

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