
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 $f(x) = 2\sin(wx + \phi) \left(w > 0, |\phi| < \frac{\pi}{2}\right)$ has a minimum positive period of $\pi$, and its graph passes through the point $(\frac{\pi}{3}, 2)$.

(1) Find the analytic expression of $f(x)$ and its monotonically increasing interval.

(2) Find the maximum and minimum values of $f(x)$ when $x \in [0, \frac{\pi}{2}]$ and the corresponding $x$ values when these extremes are reached.

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