
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
Let the sides opposite to the internal angles $A$, $B$, and $C$ of $\triangle ABC$ be $a$, $b$, and $c$ respectively, and let $\overrightarrow{u}=(b,-\sqrt{3}a)$, $\overrightarrow{v}=(\sin A,\cos B)$, and $\overrightarrow{u}\perp \overrightarrow{v}$.  
$(1)$ Find the magnitude of angle $B$;  
$(2)$ If $b=3$ and $c=2a$, find the values of $a$ and $c$.

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