
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 $f(x)=2\sin(x)\cos\left(x-\frac{\pi}{3}\)-\frac{\sqrt{3}}{2}$. In acute triangle $\triangle ABC$, with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$, respectively, and the radius $R$ of the circumcircle of $\triangle ABC$, given that $a\cos(B)-b\cos(A)=R$.  
$(1)$ If $f(A)=1$, find $B$;  
$(2)$ Find the range of $\frac{R-c}{b}$.

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