
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 fractional equation about $x$: $\frac{4}{x+1}+\frac{3}{x-1}=\frac{k}{{{x}^{2}}-1}$<br/>$(1)$ If the equation has a positive root, find the value of $k$;<br/>$(2)$ If the solution of the equation is a negative number, find the range of values for $k$.

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