
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 the function $f(x)= \sqrt {3}\sin x\cdot\cos x- \frac {1}{2}\cos 2x$ $(x\in\mathbb{R})$.  
$(1)$ Find the minimum value and the smallest positive period of the function $f(x)$.  
$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $f(C)=1$, $B=30^{\circ}$, $c=2 \sqrt {3}$. Find the area of $\triangle ABC$.

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