
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
Given the following conditions:①$asinB=bsin({A+\frac{π}{3}})$; ②$S=\frac{{\sqrt{3}}}{2}\overrightarrow{BA}•\overrightarrow{CA}$; ③$c\tan A=\left(2b-c\right)\tan C$. Choose one of the three conditions and fill in the blank below, then answer the following questions.<br/>In $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, the area of $\triangle ABC$ is $S$, and it satisfies _____.<br/>$(1)$ Find the value of angle $A$;<br/>$(2)$ Given that the area of $\triangle ABC$ is $2\sqrt{3}$, point $D$ lies on side $BC$ such that $BD=2DC$, find the minimum value of $AD$.

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