
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 vectors $\overrightarrow{a}=(\cos ( \frac {\pi}{2}+x),\sin ( \frac {\pi}{2}+x))$ and $\overrightarrow{b}=(-\sin x, \sqrt {3}\sin x)$, let $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$.  
$(1)$ Find the smallest positive period of the function $f(x)$ and the maximum value of $f(x)$.  
$(2)$ In an acute triangle $ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $f( \frac {A}{2})=1$ and $a=2 \sqrt {3}$, find the maximum value of the area of triangle $ABC$.

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