
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 the equation $kx^{2}-2(k+1)x+k-1=0$ has two distinct real roots for $x$.<br/>$(1)$ Find the range of values for $k$;<br/>$(2)$ Determine if there exists a real number $k$ such that the sum of the reciprocals of the two real roots of this equation is equal to $0$. If it exists, find the value of $k$; if not, explain the reason.

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