
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 functions $g\left(x\right)=ax^{2}-\left(a+2\right)x$ and $h\left(x\right)=\ln x$, let $f\left(x\right)=g\left(x\right)+h\left(x\right)$.  
$(1)$ When $a=1$, find the equation of the tangent line to the function $y=g\left(x\right)$ at $x=1$.  
$(2)$ When $a$ is a positive number and $1\leqslant x\leqslant e$, if $f\left(x\right)_{\min }=-2$, find the minimum value of $a$.  
$(3)$ If $\frac{f(x_1)-f(x_2)}{x_1-x_2} \gt -2$ holds for all $0 \lt x_{1} \lt x_{2}$, find the range of values for $a$.

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