
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 points $F_1(-\sqrt{2}, 0)$ and $F_2(\sqrt{2}, 0)$, the locus of point $P$ satisfying $|PF_1| - |PF_2| = 2$ is curve $E$. The line $l: y = kx - 1$ intersects curve $E$ at two distinct points $A$ and $B$:  
(Ⅰ) Find the range of values for $k$;  
(Ⅱ) If $|AB| = 2\sqrt{5}$, find the equation of line $l$.

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