
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 that curve C is the set of points for which the absolute difference of the distances to two fixed points F₁(-2, 0) and F₂(2, 0) is equal to a constant length 2a.

1. If a = $\sqrt {3}$, find the equation of curve C.
2. If line l passes through the point (0, 1) and has only one common point with curve C from part 1, find the equation of the line.
3. If a = 1, determine whether there exists a line y = kx + 2 that intersects curve C at two points A and B such that OA is perpendicular to OB. If such a line exists, find the value of k. If not, explain why.

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