
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
In $\Delta ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, and it satisfies $\cos 2C-\cos 2A=2\sin \left(\frac{\Pi }{3}+c\right)\cdot \sin \left(\frac{\Pi }{3}-C\right)$,

$(1)$ Find the value of angle $A$     $(2)$ If $a= \sqrt{3} $ and $b\geqslant a$, find the range of values for $2b-c$

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