
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 $f\left( x \right)=\sin \omega x\cos \omega x+\sqrt{3}\sin^2\omega x-\frac{\sqrt{3}}{2}\ (\omega > 0)$ with the smallest positive period of $\pi$, the graph of the function $f\left( x \right)$ is translated left by $\frac{\pi}{6}$ units and down by $\frac{1}{2}$ unit, resulting in the graph of the function $y=g\left( x \right)$.

(I) Find the interval(s) where the function $f\left( x \right)$ is increasing;

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

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