
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 $f(x)=4\sin \left( \omega x-\dfrac{\pi }{4} \right)\cos \omega x$ reaches its extremum at $x=\dfrac{\pi }{4}$, where $\omega \in \left( 0,2 \right)$.

$(1)$ Find the smallest positive period of the function $f(x)$.

$(2)$ Shift the graph of the function $f(x)$ to the left by $\dfrac{\pi }{36}$ units, then stretch the horizontal coordinates of the resulting graph by a factor of $3$ while keeping the vertical coordinates unchanged to obtain the graph of the function $y=g(x)$. If $\alpha$ is an acute angle and $g(\alpha) =\dfrac{4}{3}-\sqrt{2}$, find $\cos \alpha$.

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