
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
Observe the characteristics of the following equations and their solutions:  
① The solution to $x+\frac{2}{x}=-3$ is $x_{1}=-1$, $x_{2}=-2$.  
② The solution to $x+\frac{6}{x}=-5$ is $x_{1}=-2$, $x_{2}=-3$.  
③ The solution to $x+\frac{12}{x}=-7$ is $x_{1}=-3$, $x_{2}=-4`;  
Answer the following questions:  
$(1)$ Write an equation that fits the above characteristics as ______, with solutions ______.  
$(2)$ Based on the characteristics of these equations, write the $n$th equation as ______, with solutions ______.  
$(3)$ Using the conclusion from $(2)$, find the solutions to the equation in terms of $x$: $x+\frac{{n}^{2}+n}{x+3}=-2\left(n+2\right)$ (where $n$ is a positive integer).

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