
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 the ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) has a vertex A(0, -1), and the distance from the right focus to the line $x - y + 2\sqrt{2} = 0$ is 3.
  (1) Find the equation of the ellipse;
  (2) If the line $y = kx + m$ ($k \neq 0$) intersects the ellipse at two distinct points M and N, when $|AM| = |AN|$, find the range of the real number $m$.

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