
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 $f(x) = 4\sin(\omega x + \frac{\pi}{3})$ ($\omega > 0$) has the smallest positive period of $\pi$, and let vector $\overrightarrow{a} = (-1, f(x))$, $\overrightarrow{b} = (f(-x), 1)$, $g(x) = \overrightarrow{a} \cdot \overrightarrow{b}$.  
(1) Find the increasing interval of the function $f(x)$.  
(2) Find the maximum and minimum values of the function $g(x)$ in the interval $\left[\frac{\pi}{8}, \frac{\pi}{3}\right]$.  
(3) If $x \in [0, 2015\pi]$, find the number of real numbers $x$ that satisfy $\overrightarrow{a} \perp \overrightarrow{b}$.

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