
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
Please read the following material:<br/>Question: Given the equation $x^{2}+x-1=0$, find a quadratic equation such that its roots are twice the roots of the given equation.<br/>Solution: Let the roots of the desired equation be $y$, then $y=2x$. Substitute $x=\frac{y}{2}$ into the given equation, we get ${(\frac{y}{2})^2}+\frac{y}{2}-1=0$;<br/>Simplify, we get $y^{2}+2y-4=0$; hence the desired equation is $y^{2}+2y-4=0$. This method of finding a new equation by substituting roots of an equation is called "root substitution". Please use the "root substitution" provided in the reading material to find the new equation (requirement: express the desired equation in general form);<br/>$(1)$ Given the equation $x^{2}+3x-2=0$, find a quadratic equation such that its roots are the negative of the roots of the given equation;<br/>$(2)$ Given a quadratic equation in terms of $x$, $ax^{2}-bx+c=0\left(a\neq 0\right)$ has two unequal non-zero real roots, find a quadratic equation such that its roots are the reciprocals of the roots of the given equation.

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