
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 $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the internal angles $A$, $B$, and $C$, respectively. Given that $a=\sqrt{3}$ and $\sqrt{3}c=3\cos B+ b\sin A$.<br/>$(1)$ Find the value of angle $A$;<br/>$(2)$ Point $A$ and $D$ are on the opposite sides of line $BC$, $BD\perp BC$, and $BD=1$. Find the maximum value of $AD$.

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