
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 $f(x)=\cos^2x+ \sqrt{3}\sin (\pi-x)\cos (\pi+x)- \frac{1}{2}$  
(I) Find the decreasing interval of the function $f(x)$ in $[0,\pi]$;  
(II) In an acute triangle $\triangle ABC$, with internal angles $A$, $B$, $C$, and opposite sides $a$, $b$, $c$ respectively, given that $f(A)=-1$, $a=2$, and $b\sin C=a\sin A$, find the area of $\triangle ABC$.

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