
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 solution set of the inequality $ax^{2}-2x+a\leqslant 0$ with respect to $x$ be $S$.  
$(1)$ If $S$ has exactly one element, then the value of $a$ is ______;  
$(2)$ If $0\in S$ and $-1\notin S$, then the range of values for $a$ is ______.

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