
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: For two unequal non-zero real numbers $a$ and $b$, if the value of the fraction $\frac{{({x-a})({x-b})}}{x}$ is zero, then $x=a$ or $x=b$. Also, because $\frac{{({x-a})({x-b})}}{x}=\frac{{{x^2}-({a+b})x+ab}}{x}=x+\frac{{ab}}{x}-({a+b})$, so the equation $x+\frac{{ab}}{x}=a+b$ has two solutions, which are $x_{1}=a$ and $x_{2}=b$. Apply the above conclusion to answer the following questions:  
$(1)$ If the two solutions of the equation $x+\frac{p}{x}=q$ are $x_{1}=-2$ and $x_{2}=3$, then $p=$______, $q=$______;  
$(2)$ If the two solutions of the equation $x+\frac{{-2}}{x}=3$ are $x_{1}=a$ and $x_{2}=b$, find the value of $a^{2}+b^{2}$;  
$(3)$ If the two solutions of the equation $2x+\frac{{2{n^2}+n}}{{2x+1}}=3n$ are $x_{1}$ and $x_{2}$ $(x_{1} \lt x_{2})$, find $\frac{{2{x_1}+1}}{{2{x_2}-2}}$ (express using an algebraic expression with $n$).

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