
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
Choose one of the following three conditions, $2a\cos C=2b-c$, $\sqrt{3}bcos\frac{B+C}{2}=asinB$, $asinC=ccos({A-\frac{π}{6}})$, and add it to the following question, then solve it.  
Given an acute triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, and fill in the blank with the corresponding number.  
$(1)$ If $a=\sqrt{3}$ and $cos(B-C)=\frac{9}{10}$, find the area of $\triangle ABC$.  
$(2)$ Find the range of values for $\frac{b+c}{a}$.

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