
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 $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$ respectively, and it is given that $\sqrt {3}c\cosA=a\sinC$.
(1) If $4\sinC=c^2\sinB$, find the area of $\triangle ABC$;
(2) If $\vec{AB} \cdot \vec{AC}=4$, find the minimum value of $a$.

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