
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 that the function $f(x) = 2\sin\omega x\cos\omega x + 2\sqrt{3}\sin^2\omega x - \sqrt{3}\ (\omega > 0)$ has a minimum period of $\pi$.

(1) Find the interval where the function $f(x)$ is monotonically increasing.

(2) The graph of the function $f(x)$ is shifted to the left by $\frac{\pi}{6}$ units and up by $1$ unit to obtain the graph of the function $y = g(x)$. Find the maximum and minimum values of the function $y = g(x)$ on the interval $\left[-\frac{\pi}{12}, \frac{\pi}{3}\right]$.

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