
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 $F\_2(\sqrt{2}, 0)$ that satisfy the condition $|PF\_2| - |PF\_1| = 2$, find the trajectory of point $P$, which is curve $E$.

1. Determine 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}
user