
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 $a$, $b$, $c$ with $a > b$, $a > c$. The radius of the circumcircle of $\triangle ABC$ is $1$, and the area of $\triangle ABC$ is $S= \sqrt{3} \sin B\sin C$.

$(1)$ Find the size of angle $A$;

$(2)$ If a point $D$ on side $BC$ satisfies $BD=2DC$, and $AB\perp AD$, find the area of $\triangle ABC$.

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