
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 in $\Delta ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, and $\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{2\sqrt{3}\sin A}{3\sin C}$.

$(1)$ Find the value of $b$;

$(2)$ If $\cos B+\sqrt{3}\sin B=2$, find the range of values for $a+c$.

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