
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 $f\left(x\right)=ax^{2}-\left(a+2\right)x+\ln x$.  
$(Ⅰ)$ When $a=1$, find the slope of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(1,f\left(1\right)\right)$.  
$(Ⅱ)$ When $a \gt 0$, the minimum value of the function $f\left(x\right)$ on the interval $\left[1,e\right]$ is $-2$. Find the range of values for $a$.  
$(Ⅲ)$ If for any $x_{1}$, $x_{2}\in \left(0,+\infty \right)$, where $x_{1} \lt x_{2}$, and $f(x_{1})+2x_{1} \lt f(x_{2})+2x_{2}$ always holds, find the range of values for $a$.

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