
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 $f(x) = A\sin(\omega x + \alpha)$ where $A > 0$, $\omega > 0$, and $-\frac{\pi}{2} < \alpha < \frac{\pi}{2}$. The smallest positive period of the function is $\pi$, and $f(x)$ reaches its maximum value of 3 at $x = \frac{\pi}{6}$.

(I) Find the analytical expression of $f(x)$ and its interval of increase.

(II) If $x_0 \in [0, 2\pi)$ and $f(x_0) = \frac{3}{2}$, find $x_0$.

(III) After shifting the graph of the function $f(x)$ to the right by $m (m > 0)$ units to obtain the graph of $y = g(x)$, and if $y = g(x)$ is an even function, find the minimum value of $m$.

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