
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 an acute-angled $\triangle ABC$ with the opposite sides of angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, and $(a^2+b^2-c^2)\sin C=\sqrt{3}ab\cos C$.

$(1)$ Find angle $C$;

$(2)$ If $c=\sqrt{3}$, find the range of values for $b-2a$.

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