
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 $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $c=2$, the angle bisector of $\angle BAC$, $AD$, intersects $BC$ at point $D$.  
$(1)$ If $b=1$ and $\angle BAC=60^{\circ}$, find the length of $AD$.  
$(2)$ If $\triangle ABC$ is an acute triangle and $\frac{2a}{b}=1+\frac{\tan C}{\tan B}$, find the range of values for the perimeter of $\triangle ABC$.

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