
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 $f(x)=\sqrt{3}\sin \omega x\cos \omega x-\sin^{2} \omega x+1(\omega > 0)$, the distance between adjacent symmetry axes is $\frac{\pi }{2}$. (I) Find the value of $\omega$ and the monotonically decreasing interval of the function $f(x)$;

(II) Given that $a,b,c$ are the sides opposite to angles $A,B,C$ respectively in $\Delta ABC$, and satisfy $b=2,f(A)=1$, $2\sin A=\sqrt{3}\sin C$, find the area of $\Delta ABC$.

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