
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 triangle $ABC$ with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$, respectively, such that $\cos \frac{A}{2} = \frac{2\sqrt{5}}{5}$ and $\overrightarrow{AB} \cdot \overrightarrow{AC} = 3$.

1. Find the area of triangle $ABC$.
2. Find the minimum value of $a$.

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