
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
Read the following material: Find the solutions of the fractional equation $x+\frac{k}{x}=m$. Let $k=ab$ and $m=a+b$. It can be obtained that $x_{1}=a$ and $x_{2}=b$ are the solutions of this fractional equation. For example, to find the solutions of the fractional equation $x+\frac{6}{x}=-5$, it can be observed that $k=6=(-2)\times(-3)$ and $m=-5=(-2)+(-3)$. It can be easily verified that $x_{1}=-2$ and $x_{2}=-3$ are the solutions of this equation. Answer the following questions based on the above material:<br/>$(1)$ Find the solutions of the fractional equation $x+\frac{5}{x}=-6$;<br/>$(2)$ If $x_{1}=m$ and $x_{2}=n$ are two solutions of the fractional equation $x-\frac{3}{x}=4$, find the value of $\frac{1}{m}+\frac{1}{n}$;<br/>$(3)$ Let $a$ be a constant and $a\neq 0$. If the two solutions of the fractional equation $x+\frac{{a}^{2}+2a}{x+1}=2a+1$ are $x_{1}$ and $x_{2}$, find the value of ${({x}_{1}-{x}_{2})}^{2}$.

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