
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 $f(x)=\sin (2x- \frac {\pi}{6})+2\cos ^{2}x-1$.  
(I) Find the intervals of increase for the function $f(x)$.  
(II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ respectively, and $a=1$, $b+c=2$, $f(A)= \frac {1}{2}$, find the area of $\triangle ABC$.

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