
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 angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, and it satisfies the condition 
$$\frac{c}{\sin C} = \frac{a}{\sqrt{3}\cos A}$$
(1) If $4\sin C = c^2 \sin B$, find the area of triangle $ABC$;
(2) If $\vec{AB}\cdot\vec{BC} + \|\vec{AB}\|^2 = 4$, find the minimum value of $a$.

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