
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 that an ellipse has a vertex at $A(0,-1)$ and its foci are located 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 standard equation of the ellipse;
$(II)$ Suppose there is a line $l: y = x + m$; is there a real number $m$ such that line $l$ intersects the ellipse from part $(1)$ at two distinct points $M$ and $N$, and $|AM| = |AN|$? If such an $m$ exists, find its value; if not, explain why.

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