
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 the condition $|PF_{1}|-|PF_{2}|=2$ is curve $E$. The line $l: y=kx-1$ intersects curve $E$ at two distinct points $A$ and $B$:

$(1)$ Find the range of values for $k$;

$(2)$ If $|AB|=2\sqrt{5}$, find the equation of line $l$.

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