
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 with one vertex at $A(0, -1)$ and its foci on the $x$-axis. If the distance from the right focus to the line $x-y+2\sqrt{2}=0$ is $3$.

(Ⅰ) Find the equation of the ellipse;

(Ⅱ) Suppose the ellipse intersects the line $y=kx+m \ (k\ne 0)$ at two distinct points $M$ and $N$. When $|AM|=|AN|$, find the range of values for $m$.

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