
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
The famous mathematician Gauss once said: "If others think deeply and persistently about mathematical truths like I do, they will also find my discoveries." Let us emulate this great man and apply the spirit of diligent thinking, good learning, and steadfast practice to our daily mathematical studies, trying to discover new surprises.  
**[Problem Statement]**  
We have previously explored the relationship between the roots and coefficients of a quadratic equation. If the coefficients of a quadratic equation change according to a certain rule, will there be a connection between the roots of the original equation and the roots of the new equation?  
**[Establishing Relationship]**  
By enlarging or reducing the quadratic term coefficient, linear term coefficient, and constant term of a quadratic equation in the ratio of $n:1:\frac{1}{n}$, where $n\neq 0$, we refer to the new equation as the "coefficient-variable equation" of the original equation, with the coefficient variation factor being $n$.  
$(1)$ When the coefficient variation factor is $3$, find the "coefficient-variable equation" of the quadratic equation $x^{2}+2x-3=0$.  
**[Exploration on Own]**  
$(2)$ Given a quadratic equation with two real roots $x_{1}$ and $x_{2}$, when $n=2$, the "coefficient-variable equation" also has two real roots $p$ and $q$, and $x_{1}x_{2}=1$. Find the minimum value of $\frac{q}{p}+\frac{p}{q}-(\frac{4}{p}+\frac{1}{4q})+17$.  
$(3)$ Given the equation in terms of $x$: $(3x^{2}+tx-2)^{2}+(-2x^{2}-tx+3)^{2}=(x^{2}+1)^{2}$ has four real roots $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$. Is there a constant value $k$ such that for any real number $t$, $\frac{{x}_{1}}{{x}_{2}}=\frac{{x}_{3}}{{x}_{4}}=k$? If it exists, find the value of $k$; if not, explain the reason.

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