
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 $\triangle ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, if $f(A)=0$, $\sin B= \frac {4}{5}$, $a= \sqrt {3}$, find the value of $b$.

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