
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 $f(x)=ax^{2}-2x+b(a\neq 0)$ attains a minimum value of $0$ at $x=1$.  
$(1)$ Find the values of $a$ and $b$.  
$(2)$ Let $g(x)=\frac{f(x)}{x}$. Find the minimum and maximum values of the function $y=g(2^{x}-1)$ for $x∈[\frac{1}{2},2]$, and determine the corresponding values of $x$ when the minimum and maximum values are attained.

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