
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 acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=2$ and $\frac{a}{b}=\frac{\sqrt{3}}{3}\sin C+\cos C$.

$(1)$ Find angle $B$.

$(2)$ If $M$ is a moving point inside $\triangle ABC$ and satisfies $\overrightarrow{BM}=\overrightarrow{MA}+\overrightarrow{MC}$, does $|\overrightarrow{BM}|$ have a maximum value? If it does, find the maximum value and the conditions under which it occurs. If not, explain why.

$(3)$ If $D$ is a point on $AC$ in $\triangle ABC$ and satisfies $\frac{\overrightarrow{BA}\cdot\overrightarrow{BD}}{|\overrightarrow{BA}|}=\frac{\overrightarrow{BD}\cdot\overrightarrow{BC}}{|\overrightarrow{BC}|}$, find the range of values for $AD:DC$.

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