
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 a quadratic function $y=ax^{2}+bx-4$ ($a$ and $b$ are constants, and $a\neq 0$) whose graph passes through the point $(3,-1)$.  
$(1)$ Determine whether the point $(2,2-2a)$ lies on the graph of this function, and explain the reason.  
$(2)$ If the graph of this quadratic function intersects the $x$-axis at only one point, find the expression of the function.  
$(3)$ Given that the graph of the quadratic function passes through the points $(x_{1},y_{1})$ and $(x_{2},y_{2})$, and when $x_{1} \lt x_{2}\leqslant \frac{2}{3}$, it always holds that $y_{1} \gt y_{2}$, find the range of values for $a$.

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