
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:<br/>For two distinct 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 $\frac{{(x-a)(x-b)}}{x}=\frac{{{x^2}-(a+b)x+ab}}{x}=x+\frac{{ab}}{x}-\left(a+b\right)$, so the equation $x+\frac{{ab}}{x}=a+b$ has two solutions for $x$, which are $x_{1}=a$ and $x_{2}=b$. Apply the above conclusion to answer the following questions:<br/>$(1)$ If the two solutions of the equation $x+\frac{p}{x}=q$ are $x_{1}=-2$ and $x_{2}=3$, then $p=$______, $q=$______; <br/>$(2)$ The larger of the two solutions of the equation $x+\frac{7}{x}=8$ is ______; <br/>$(3)$ If the two solutions of the equation $2x+\frac{{{n^2}-n}}{{2x-1}}=2n$ are $x_{1}$ and $x_{2}(x_{1} \lt x_{2})$, find the value of $\frac{{2{x_1}-1}}{{2{x_2}}}$ in terms of the variable $n$.

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