
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 triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. It is given that $a = b\cos C + \frac{\sqrt{3}}{3}c\sin B$.

$(1)$ Find the angle $B$;

$(2)$ If $a = 2$ and the area of $\triangle ABC$ is $2\sqrt{3}$, find the value of side $b$.

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