
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
Let vector $\overrightarrow{a}=(\sin x, \frac {\sqrt {3}}{2}(\sin x-\cos x))$, $\overrightarrow{b}=(\cos x,\sin x+\cos x)$, where $x\in\mathbb{R}$, and denote the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$.  
$(1)$ Find the intervals of monotonic increase for the function $f(x)$.  
$(2)$ In an acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. If $f(A)= \frac {1}{2}$ and $a= \sqrt {2}$, find the maximum area of $\triangle ABC$.

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