
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 $f\left(x\right)= \sqrt{3}\sin x\cos x+{\cos }^{2}x+ \frac{3}{2}$.

(1) Find the range of the function $y=f(x)$ when $x\in\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$.

(2) Given $\omega > 0$, the function $g\left(x\right)=f\left( \frac{\omega x}{2}+ \frac{\pi}{12}\right)$. If the smallest positive period of the function $g(x)$ is $\pi$, find the value of $\omega$ and the increasing interval of the function $g(x)$.

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