
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 angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $bcos\frac{A}{2}=asinB$.<br/>$(1)$ Find angle $A$;<br/>$(2)$ Let $a=6$, $O$ be a point inside $\triangle ABC$, the extension of segment $AO$ intersects $BC$ at point $D$, ____, find the area of $\triangle ABC$.<br/>Choose one of the following conditions to supplement the blank line below to ensure the existence of $\triangle ABC$ and solve the problem.<br/>① $O$ is the centroid of $\triangle ABC$, $OA=2\sqrt{3}$;<br/>② $O$ is the circumcenter of $\triangle ABC$, $OA=4$;<br/>③ $O$ is the incenter of $\triangle ABC$, $AD=3\sqrt{3}$.<br/>Note: If multiple conditions are chosen to answer separately, points will be awarded based on the first answer. If the selected condition number cannot form a triangle, the reason must be explained, and no need to choose another number to answer.

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