
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 an acute triangle $\triangle ABC$ where the internal angles $A$, $B$, and $C$ are opposite to sides $a$, $b$, and $c$ respectively, and $a=3$, $b=\sqrt{13}$. If $a\sin 2B=b\sin A$. Find:<br/>
$(1)$ the value of angle $B$;<br/>
$(2)$ if point $D$ satisfies $\overrightarrow{BD}=\frac{1}{3}\overrightarrow{BA}+\frac{2}{3}\overrightarrow{BC}$, find the length of $BD$.

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