
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 $f(x) = \cos\omega x \cdot \sin\omega x + \sqrt{3} \cos^2\omega x - \frac{\sqrt{3}}{2}$ where $0 < \omega \leq 1$, and it satisfies $f(x+\pi) = f(x)$,  
(I) Find the explicit formula for $y = f(x)$.  
(II) Determine the range of $y = f(x)$ when $x \in \left[-\frac{\pi}{12}, \frac{5\pi}{12}\right]$.  
(III) If $3f(x)^2 + mf(x) - 1 = g(x)$, find the minimum value of $g(x)$ on the interval $x \in \left[-\frac{\pi}{12}, \frac{5\pi}{12}\right]$.

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