
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 $f(x) = 4\sin(\omega x - \frac{\pi}{4}) \cdot \cos(\omega x)$ achieves its extremum at $x = \frac{\pi}{4}$, where $\omega \in (0,2)$.
(1) Find the smallest positive period of the function $f(x)$.
(2) Shift the graph of the function $f(x)$ to the left by $\frac{\pi}{36}$ units, then stretch the horizontal axis of the resulting graph by a factor of $3$, while keeping the vertical axis unchanged, to obtain the graph of the function $y = g(x)$. If $\alpha$ is an acute angle and $g(\alpha) = \frac{4}{3} - \sqrt{2}$, find $\cos \alpha$.

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