
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 $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, 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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