
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 $f(x)=\frac{1}{2}\sin 2x-\cos^{2}(x+\frac{\pi }{4})$:

(I) Find the interval(s) where $f(x)$ is monotonically increasing for $x\in R$;

(II) In $\triangle ABC$, let the sides opposite to angles $A$, $B$, $C$ be $a$, $b$, $c$ respectively. If $f(\frac{B}{2})=0$, $\angle B$ is acute, $b=1$, $c=2$, find the area of $\triangle ABC$.

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