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

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