
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 the vectors $$\overrightarrow {a}=(\sin x, \frac { \sqrt {3}}{2}(\sin x-\cos x))$$ and $$\overrightarrow {b}=(\cos x, \sin x+\cos x)$$, where $x\in\mathbb{R}$. Define 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 $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, 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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