
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 $f(x)=\sqrt{3}\cos (\frac{\pi }{2}+x)\bullet \cos x+\sin^{2}x$, where $x\in R$.
(I) Find the interval where $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $B=\frac{\pi }{4}$, $a=2$ and angle $A$ satisfies $f(A)=0$, find the area of $\triangle ABC$.

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