
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]


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In the Cartesian coordinate system $xOy$, there is a parabola $y=ax^{2}-(a+1)x$.  
$(1)$ If the parabola passes through the point $(2,0)$, find the axis of symmetry of the parabola.  
$(2)$ If $M(x_{1},y_{1})$ and $N(x_{2},y_{2})$ are two different points on the parabola.  
① When $x_{1}+x_{2}=-4$ and $y_{1}=y_{2}$, find the value of $a$.  
② If for $x_{1} \gt x_{2} \geq -2$, it holds that $y_{1} \lt y_{2}$, find the range of values for $a$.

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