
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 one of its vertices at $A(0, -1)$, and the distance from its 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 $N$, find the range of real number $m$ when $|AM| = |AN|$.

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