
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
Let the quadratic function $y=ax^{2}+bx+1$, where $a \gt 0$ and $b\in R$, have a minimum value of $-a$, and the two roots of the equation $ax^{2}+bx+1=0$ be $x_{1}$ and $x_{2}$.  
$(1)$ Find the value of $x_{1}-x_{2}$;  
$(2)$ If the solution set of the inequality $ax^{2}+bx+1 \lt 0$ with respect to $x$ is $A$, and the function $y=ax^{2}+\left(b+2\right)x+1$ does not have a minimum value in $A$, find the range of values for $a$;  
$(3)$ If $-2 \lt x_{1} \lt 0$, find the range of values for $b$.

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