
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 a moving point P $(x, y)$ on a curve $C$, satisfying that the ratio of the distance to a fixed point $A$ $(-1, 0)$ over the distance to another fixed point $B$ $(1, 0)$ is $\sqrt{2}$:
1. Find the equation of the curve $C$.
2. If a line $l$ passing through the point $M$ $(1, 2)$ intersects the curve $C$ at two points $M$ and $N$, and if $|MN| = 4$, find the equation of the line $l$.

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