
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 acute triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and $\sqrt{3}b=2asinBcosC+2csinBcosA$.  
$(1)$ Find the measure of angle $B$;  
$(2)$ Given $a=3$, $c=4$,  
① Find $b$,  
② Find the value of $\cos \left(2A+B\right)$.

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