
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 distance from a center of symmetry of the graph of $y=f(x)$ to the nearest axis of symmetry is $\frac{\pi}{4}$.

(Ⅰ) Find the value of $\omega$ and the equation of the axis of symmetry for $f(x)$;

(Ⅱ) In $\triangle ABC$, with the sides opposite angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively. If $f(A)= \frac{ \sqrt{3}}{4},\sin C= \frac{1}{3},a= \sqrt{3}$, find the value of $b$.

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