
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 $f(x) = \cos{\omega x} \cdot \sin(\omega x - \frac{\pi}{3}) + \sqrt{3}\cos^2{\omega x} - \frac{\sqrt{3}}{4} (\omega > 0, x \in \mathbb{R})$, and the graph of the function $y = f(x)$ has a symmetry center whose nearest distance to its axis of symmetry is $\frac{\pi}{4}$.
$(1)$ Find the value of $\omega$ and the equation of the axis of symmetry for $f(x)$;
$(2)$ In $\triangle ABC$, where the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, if $f(A) = 0$, $\sin B = \frac{4}{5}$, and $a = \sqrt{3}$, find the value of $b$.

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