
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 $C$ with its center at the origin and an eccentricity of $\frac{\sqrt{6}}{3}$. The distance from the right focus of the ellipse to the line $x + y + \sqrt{2} = 0$ is $2$.  
$(1)$ Find the standard equation of the ellipse $C$.  
$(2)$ The lower vertex of the ellipse $C$ is $A$. The line $y = kx + m (k \neq 0)$ intersects the ellipse $C$ at two distinct points $M$ and $N$. When $|AM| = |AN|$, determine the range of values for $m$.

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