
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: $f\left(x\right)=x^{2}-|ax-b|$, (where $a\in R^{+}$, $b\in R$).<br/>$(1)$ If $a=b=1$, find the minimum value of $f\left(x\right)$;<br/>$(2)$ If $a=2$, $b\geqslant 2$, and the domain and range of the function $f\left(x\right)$ are both $\left[1,b\right]$, find the value of $b$;<br/>$(3)$ If the graph of the function $f\left(x\right)$ has $2$ distinct intersection points with the line $y=1$ for $x\in \left(0,2\right)$, find the range of $\frac{b}{a}$.

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