
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\cos \frac{A}{2}= \frac{2 \sqrt{5}}{5}, \overrightarrow{AB} \cdot \overrightarrow{AC}=15$.  
$(1)$ Find the area of $\triangle ABC$;  
$(2)$ If $\tan B=2$, find the value of $a$.

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