
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
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given $2b\cos C=2a+c$. Find:<br/>
$(1)$ Angle $B$;<br/>
$(2)$ Let $b=9$. If point $M$ is a point on side $AC$ such that $2\overrightarrow{AM}=\overrightarrow{MC}$ and $\angle MAB=\angle MBA$, find the area of $\triangle BMC$.

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