
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
Given the function $y=2x^{2}-(a+2)x+a$, where $a\in R$.  
$(1)$ When $a=-1$, find the solution to the inequality $y \gt 0$ in terms of $x$.  
$(2)$ If the equation $2x^{2}-(a+2)x+a=x+1$ has two positive real roots $x_{1}$ and $x_{2}$, find the minimum value of $\frac{x_{2}}{x_{1}}+\frac{x_{1}}{x_{2}}$.

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