
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 $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $a=6$ and $b\sin\frac{B+C}{2}=a\sin B$. Find:
1. The measure of angle $A$.
2. Let $M$ be a point inside triangle $\triangle ABC$. Extend $AM$ to intersect $BC$ at point $D$. _______. Find the area of triangle $\triangle ABC$. Choose one of the following conditions to supplement the blank line to ensure the existence of triangle $\triangle ABC$ and solve the problem:
   - $M$ is the circumcenter of triangle $\triangle ABC$ and $AM=4$.
   - $M$ is the centroid of triangle $\triangle ABC$ and $AM=2\sqrt{3}$.
   - $M$ is the incenter of triangle $\triangle ABC$ and $AD=3\sqrt{3}$.
   (Note: The point of intersection of the perpendicular bisectors of the sides of a triangle is called the circumcenter, the point of intersection of the medians is called the centroid, and the point of intersection of the angle bisectors is called the incenter.)

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