
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 two fixed points $F_1(-\sqrt{2}, 0)$ and $F_2(\sqrt{2}, 0)$, the locus of point $P$ satisfying the condition $|PF_2| - |PF_1| = 2$ is curve $E$.
(1) Find the equation of curve $E$;
(2) Suppose a line passing through the point $(0, -1)$ intersects curve $E$ at points $A$ and $B$. If $|AB| = 6\sqrt{3}$, find the equation of line $AB$.

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