
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
We have learned that $\left(x-a\right)\left(x-b\right)=x^{2}-\left(a+b\right)x+ab$. If a fractional equation in $x$ satisfies $x+\frac{{ab}}{x}=a+b$ (where $a$ and $b$ are non-zero integers), and the equation has two roots $x_{1}=a$ and $x_{2}=b$, we call such an equation a "cross equation". For example, $x+\frac{2}{x}=3$ can be transformed into $x+\frac{{1×2}}{x}=1+2=3$, so $x_{1}=1$ and $x_{2}=2$. Another example is $x+\frac{6}{x}=-5$, which can be transformed into $x+\frac{{({-2})×({-3})}}{x}=-2-3=-5$, so $x_{1}=-2$ and $x_{2}=-3$. Apply the above conclusion to answer the following questions:  
$(1)$ For the "cross equation" $x+\frac{8}{x}=-6$, $x_{1}=$______, $x_{2}=$______;  
$(2)$ If the two roots of the "cross equation" $x-\frac{2}{x}=-1$ are $x_{1}=a$ and $x_{2}=b$, find the value of $\frac{1}{a}+\frac{1}{b}$;  
$(3)$ If the two roots of the "cross equation" $x+\frac{{{n^2}+n}}{{x-3}}=2n+4$ are $x_{1}$ and $x_{2}$ ($x_{1} \lt x_{2}$), find the value of $\frac{{{x_2}}}{{{x_1}+1}}$.

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